The main way to win the game is to reduce your opponent's life total to 0. Sometimes there are instant and sorcery cards that do that, but usually you'll accomplish it by attacking with your creatures.
You get a chance to attack in every turn, which is known as your "combat phase". The way each turn goes is that you first untap all your permanents, then you draw your card for the turn. (If anything refers to your "upkeep", that means after you untap and before you draw.) Then you get a chance to play lands and cast spells, known as your "main phase". Then you have your combat phase, then another main phase after that, and then the turn ends. So you can play lands and cast spells either before or after combat, but not during. (Except instant spells; those can be cast at any time.)
The first thing you do is choose which of your creatures are going to attack. You can attack with any number of creatures, but it has to all be at the same time; you don't get to try attacking with one and see how it goes, then try another. A creature has to be untapped in order to attack; tapped creatures have been "used", and can't attack. You also can't attack with a creature that you just played this turn; creatures that enter the battlefield are considered to be "summoning sick", and you can't attack with them until your next turn.
Once you've chosen what creatures you're attacking with, you tap those creatures to indicate that they're attacking. Then your opponent gets a chance to block. Each of their creatures can block up to one of yours. They don't have to block, but it's often a good idea. Blockers also must be untapped in order to block, so if you attacked with a creature on your turn, you won't be able to use it to block on your opponent's next turn.
Now it's time for damage. Each creature deals damage equal to its power; the first number (before the slash) in the little box at the bottom right. Attacking creatures damage the creature that's blocking them if there is one, or if they weren't blocked, they damage the opponent directly. Blocking creatures damage the attacking creature that they're blocking.
(If multiple creatures blocked the same attacker, the attacking player gets to choose what order to damage the blockers in.)
Any creature that's dealt at least as much damage as its toughness (the number after the slash in the bottom right) dies, and is put into the graveyard. You don't have to worry about damage "stacking up" on a creature across multiple turns; if a creature survives a turn, all damage is removed from it when the turn ends.
Sometimes your opponent might have a planeswalker or battle on the battlefield. In this case, your creatures can attack that card instead of attacking the opponent directly. For each of your attacking creatures, you choose what they're attacking as they attack. The opponent can then block them as normal, and if unblocked, they deal damage to the planeswalker or battle instead of the opponent.
Combat is what most of the game is focused around, so many card abilities reference parts of combat. For example, a card might have a triggered ability that says "whenever a creature attacks, draw a card". The steps of combat happen one at a time, so with a trigger like this, you'd draw the card before blockers are declared.
Many creatures have "keyword abilities" that affect combat. A keyword ability is a single work or phrase that stands for something a little more complicated. There are a lot of these, so don't worry about memorizing them all right away. Just deal with them as they come up in your games. Here are some of the most common ones:
- Flying means that this creature can't be blocked except by other creatures with flying or reach.
- Reach means that this creature can block creatures with flying, despite not flying itself.
- Haste means that a creature can attack right away on the turn it's played, ignoring "summoning sickness".
- Deathtouch means that any amount of damage the creature deals to another creature is lethal. For example if a 1/1 creature with deathtouch attacks and is blocked by a 2/2, the 2/2 will die despite having only been dealt 1 damage.
- Trample means that a blocked creature gets to deal extra damage to the defending player. For example, if a 3/3 creature with trample attacks and is blocked by a 2/2, the attacker gets to kill the blocker and deal 1 damage to the defending player.
- Defender means that a creature can't attack; it's a drawback.
- First strike means that a creature deals its damage before all the other creatures. For example, if a 2/2 attacker is blocked by a 2/2 blocker, normally they'd both die, but if the attacker has first strike, then the blocker dies before it has a chance to deal damage to the attacker.
- Double strike means the same thing as first strike, but then the creature also deals its normal damage. So a 3/3 attacker with double strike that goes unblocked deals a total of 6 damage to the other player.
- Indestructible means that the creature can't be destroyed; lethal damage dealt to it will just be ignored, it stays alive.
- Lifelink means that any damage the creature deals also gains its controller that much life.
- Vigilance means that the creature doesn't have to tap to attack, so it will be able to block on the next turn too.
- Prowess means that whenever you cast an instant or sorcery spell, the creature gets +1/+1 to its power and toughness until the turn ends.
And that's only some of them! Magic is an extremely deep game with over 30 years worth of new expansions and mechanics. As you play, you'll encounter many abilities you've never seen before. Often these mechanics will be explained on the card, but sometimes they won't be. In those cases, you can look up what the keyword does in the rules glossary. If you still have questions, head over to the MTG judge chat and they'll be happy to answer. Have fun!