Magic is a game of hidden information. I'm sure you've heard that before, but it is imperative to pick up as much information as possible from what your opponent does. Sadly, that information is only useful if you know how to use it.
It's also important not to give away too much information yourself. Eventually, you can learn to tell a story with your actions that might throw your opponent off.
Magic is a difficult game, and if you want to get better, there is a lot of research you are going to have to do before you even play a game.
Are you playing Limited? I hope you know all the instant-speed spells in the format, especially combat tricks, counterspells and flash creatures. If you want to go the extra mile, other important types of cards to learn are creatures with haste and sorcery-speed removal spells.
If you are playing Constructed, knowing the major archetypes and what is in them is invaluable. It's going to be really tough to figure out what they might have if you need to think about every card in the format as opposed to just every card that is generally played in their archetype. Before you can start to infer the things your opponent could be doing, you need to know everything they might have at their disposal.
Early in a match, your first information goal should be to figure out what deck your opponent is playing. Their early land drops will help in this endeavor. There are some formats where you can tell what deck your opponent is playing by only one land. In Limited, this is tougher, but less important. In Constructed, it is imperative.
If you play online, MTGGoldfish is an invaluable tool. Go to their metagame analysis page and pick your format. This will help you quickly decipher what deck your opponent is playing. Without knowing what deck your opponent is playing, you can't begin to form a game plan, so time is of the essence.
Once you know what they could have, it's time to figure out what they do have.
First, narrow down what they could have to just what they can cast. You don't need to worry about cards that aren't in their colors, or cost more than they can afford, so don't waste your mental bandwidth.
Now that you've narrowed it down to only a few cards, how have they played up to this point? Has your opponent had an opportunity to play their removal spell and passed on it? Depending on how juicy of a target you had, they probably don't have it. This goes double for combat tricks, as people tend to play their combat tricks at the first opportunity. Have they tapped out every turn? They probably have a strong instant they are trying to get you with.
When your opponent doesn't do anything on their turn with their mana, and still has cards in their hand, it is very likely they are planning to do something on your turn. The more cards they have in their hand, the more likely they are to have the specific card that really punishes you.
What point in the game you are in, combined with the style of deck they are playing, can help you guess whether they almost certainly have something to do on your turn, or are just off-curve. Control decks aren't necessarily going to have cheap creatures, so you don't learn much when they don't play an early threat. Aggro decks that miss their two-drops almost certainly have a removal spell, even if it isn't playable on turn two. They wouldn't have kept their hand without a two-drop or early interaction.
Turn five they leave up all their mana and have a full grip? They definitely have something, regardless of archetype. If they are ahead, it's probably a counterspell or a removal spell; if they are behind, the chances of them having a flash creature go up.
Here is an example of a mistake I made, because I didn't use the information my opponent was giving me.
Here I am casting Titanic Growth on my unblocked Deathbloom Thallid, trying to end the game with this attack.
My opponent had just played the Capture Sphere on my Blood Glutton during my turn, instead of on their own turn when I was tapped out, which would have let them deal an extra 2 damage with their two Mistral Singer. This should have set off alarm bells in my head. Instead I thought, "I don't think they have Unsubstantiate here, and even if they do it's not awful. I can just replay the Deathbloom Thallid." The way they played 100% says they have something. They could've blocked my Thallid if they didn't have anything, and that alone would've been a good enough reason to wait on their Capture Sphere.
Once they blocked my 3/3 Drowsing Tyrannodon with a 2/3 Jeskai Elder they clearly had something. But somehow I got it in my head that I could safely go for the win here, even though I was in no rush to close the game. The only way I was losing was to get three-for-one'd, losing my tempo and card advantage, and then my opponent drawing well.
Welp, they had Swift Response for my Deathbloom Thallid and drew well, so I lost. If I had just accepted that my opponent was doing things with a reason, I could've sniffed out a removal spell there.
Using the information you've gleaned from your opponent can be tricky. The most common things to play around are combat tricks, counterspells, removal, flash creatures, bombs and wrath effects. Each has its own quirks and ways to beat it. Sometimes you have to force them to have it, sometimes you need to try to get them to "waste" their mana by giving them nothing to do with it this turn. Most of this comes down to game plan and role assessment, but there are finer points as well.
With combat tricks, I usually like to force them to have it.
If you are on defense and they attack with open mana, they probably have something, and it's usually a good idea to make them use their trick if your creature is at all expendable. If your blocker is key to your game plan, like Goblin Rabblemaster or Selesnya Guildmage, letting them bluff you is usually ok. If you have a Centaur Courser and they attack with a Grizzly Bears, make them use the trick, as playing around it for the rest of the game is going to be harder playing without your vanilla 3/3. On top of that, it probably uses their whole turn so you don't even fall behind on tempo in most cases.
If you are on offense, and your opponent leaves up mana, it might be right to just advance your board and force them to waste their mana. Again, if your creature is worse than the trick, you might want to get them to use it, but this is often a good spot to just advance the board and force them to fall further behind. If they are blue (and to a lesser extent black), they may have instant-speed card draw—so forcing them to use their trick, which they are even less likely to have, is a good choice because they are likely to use their mana anyway.
Once you think they have a counterspell, you usually want to play your weaker threats first. This is often called bait, and works best when you can double spell. Sometimes, playing nothing and passing is best for dealing with counterspells, but you have to weigh that against the chance they have card draw and can put you far behind.
If you only have one thing to do, I usually like to force their hand and make them have it. Sitting back for too long is how you lose to controlling decks, but if your card is strong enough that it will win on its own, waiting till you can force them to play their counterspell is usually a good idea.
Removal is tricky because their goal is to spend it on the most important threat, so you need to figure out a way to get them to spend it on the wrong thing. You also don't want to get blown out by double blocking into open mana, as either a removal spell or a combat trick can punish you. Having a combat trick against red or green removal spells can often help you turn their blowout on its head, and punish them.
Against removal spells when blocking, I like to overblock. Blocking with significantly more damage than lethal, often with every creature except the most important ones, will keep you from getting really blown out by a removal spell, and can keep their two-for-one from also gaining them a tempo advantage.
This is a position from a draft I was in recently, just before drawing Crash Through for turn:
If I want to deal the most possible damage, I could play Crash Through, then Dub on my Anointed Chorister and attack for 9, leaving my opponent dead next turn even if they have a creature. Is this the right choice?
It's not, and I don't think it's particularly close either. My opponent had five cards in hand and four mana untapped. They had also played multiple instant-speed removal spells game one. To top it all off, they had missed land drops until this turn, so I knew their hand was all spells. All of this essentially telegraphed the Capture Sphere.
Before doing anything else, we are definitely supposed to attack to see what our opponent is going to do. If they do nothing, they are dead and this game is over unless they have Baneslayer Angel. If they play Capture Sphere, we can play Goblin Wizardry, and set up for an explosive amount of damage on the next turn. There is also an argument for playing the Crash Through to try to draw into a land, threatening a two-turn clock with removal up even if we don't draw a land. I think that play is not good enough, though, as the Crash Through plus Soul Sear the turn after we play Goblin Wizardry would be just as good, and doesn't require us to draw a land.
If you expect a flash creature, it is usually correct not to attack at all without a combat trick or removal spell in reserve. The most common time this is the case is when they have five mana open. Anything less, and they will likely only be able to trade with anything but your smallest creatures.
At five mana, the creatures that WotC gives flash are often large enough to eat your creature and survive combat, though it is very helpful to know what the flash threats are in the format and what you need to play around specifically. Force them to use their mana at the end of your turn, and not get the extra value of blowing you out.
If you know your opponent has a bomb in their deck, or a threat that you cannot beat if left unchecked, save any removal spell you have that can answer it, even if it means taking a strategic hit elsewhere.
Try to use your creatures as removal when possible. Sometimes just taking a few hits from a creature while you get enough resources to deal with it another way is the best choice. Attempt to apply pressure and get your opponent to play their bomb as soon as possible so you can answer it and be back to square one, able to use your spells as you best see fit in the moment.
Wrath effects kind of go in a similar vein to bombs because they are game ending. Try not to play all of your creatures if you don't have to. Sometimes your opponent just attacks you with their last creature, leaving themselves otherwise dead on board. This is the most common wrath "tell," and you should block, even if you have to trade an important creature and lose lethal. This goes double if you know they have a wrath in their deck.
"Telegraphing," or practically telling your opponent what you have in your hand by the way you play, is always detrimental. If your opponent can easily assume you have a certain card, or even a certain subset of cards, it makes their decisions much easier.
The best way to prevent this is to represent as many different realistic and relevant things possible. Flash decks are very good at this, which is why they are so popular. When your opponent has to play around a counterspell, a flash creature, a removal spell and card draw every turn, it is extremely unlikely they will be able to make the right play. If you can make your opponent believe that you have something you don't, even better.
In order to deceive your opponent successfully, first you must evaluate what parts of the game you can fall behind on and still win. If you can't afford to get behind on board, leaving up mana to represent a counterspell instead of playing your creature is just going to leave you behind on what is important in the matchup. Inversely, in slower controlling matchups, leaving up counterspells and practically playing like you have an upkeep cost of counterspell mana every turn can be very effective.
Two examples of this I can think of are the Caw-Blade mirror and Pauper Faeries. In the Caw-Blade mirror, it was often correct to not play Stoneforge Mystic on turn two and just represent mana leak, whether you had it or not. This practically screams that you have the Mana Leak, so your opponent will have to play around it. In Pauper Faeries, you often want to pass with two mana up starting as early turn two, even when you have a Preordain or Ponder. This forces your opponent to respect that you might have Spellstutter Sprite or Counterspell, putting them in a tight bind moving forward.
Once you understand that your opponent is skilled enough to suss out what you're doing, you can start to do some very crafty things. This game between two Hall of Famers shows an incredible level of planning by Antoine Ruel, and is one of my favorite examples of leveling and storytelling.
Let's start by saying none of the early turns of this game were conventional. The turn-one tapped land from Antoine, not spending his Force Spike on Kenji's Mana Leak, and following it up by Force Spike the turn-three Psychatog from Kenji were all wild. But all of these plays were made with good reasons backing them.
Kenji believed that Antoine didn't have a Force Spike from how the game was played up until this point, and rightfully so. Antoine intentionally made a very compelling argument that he did not draw it, by playing in a way that most players wouldn't have.
In most games of Magic, all things being equal, the player who makes the least mistakes will win. Making an effort to decipher what your opponent has in their hand will help you make fewer mistakes in the long run. Similarly, playing in a way to disguise what you have will cause your opponents to make more mistakes. As always, practice goes a long way toward honing these skills, but eventually it gets easier to recognize the common situations that occur and what they mean.
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All of this should be taken not as gospel, but as a baseline. Magic is a very complicated game, and there is no perfect way to play. But once you understand what is "usually" correct, you can begin to find the times where veering from the norm is profitable. I don't mean to suggest that there is a best way to decipher or play around specific cards or strategies, but to provide a foundation for you to build off of and improve upon.