BassChan said: "All right, let's see how much I can summarize here...
Before the game begins, each player shuffles and cuts their opponent's libraries(decks). They then decide who goes first, draw seven cards, and check if the hands can be kept. Now, if one player decides to mulligan, they reshuffle and draw a hand with one less card than the previous. If they had priority, they lose it, and must yield the choice of "who goes first" to the other player.
Turns occur in several steps. The steps are as follows:
Untap Step: The active player untaps their permanents, or turns all permanents they control facing "normally."
Upkeep Step: Certain effects happen here, i.e. cumulative upkeep or "At the beginning of your upkeep" effects.
Draw Step: The active player draws a card.
Pre-Combat Main Step: The active player can play a land, summon creatures, cast sorceries or instants, play artifacts, et cetera. Essentially, set up.
Combat Step: Assign your creatures to battle! Creatures summoned during the turn may not be able to attack, unless they have "Haste." It's a bit of an effect called Summoning Sickness, but we all know that. Though, warmed-up creatures may charge into battle and smack the opponent! The combat step is actually split up into it's own set of steps for "fairness."
> Assign Attackers: Assign the creatures you want to attack this turn. Turn them sideways(tap them) to show they are attacking, lest they have "Vigilance"(attacking does not cause this creature to tap). You can also throw some instant spells out to beef them up if you want.
> Assign Blockers: Your opponent can assign blocking creatures(untapped) to stop them from taking damage. Your opponent, as well, can throw some instants down, even shoot some of your stuff with a Shock or something. You can do the same to theirs.
> Assign Combat Damage: Creatures that are blocked will NOT do damage to your opponent, lest they have "Trample." Creatures caught in combat with each other will deal damage to each other at the same time, unless one possesses "First Strike," in which it will hit first. Damage is dealt to the "Toughness" part(2/3: 2 is the Power, 3 is the Toughness). If a creature's Toughness is reduced to 0, it dies, unless it's Indestructible or someone regenerates it.
And, that's really it for combat.
Post-Combat Main Step: Think of this as a continuation of your first Main Step. Didn't play a land? Go ahead and smack one down. Have a new creature to summon, now that combat is done? Bring it out. Yadda, yadda.
End Step: End of the turn crap happens. Mana pools empty, so that any unspent mana will come back and bite a player for 1 damage in each instance of 1 mana. Players can cast instant spells here, as well. Once the End Step is done, it is the next person's turn, and any damage existing on creatures goes away(A damaged 2/3 that was a 2/1 after combat becomes a 2/3 again).

That's the gist of it. Everything else is really specific crap that is explained when the time is right, like Planeswalkers(which are still new to me) and abilities like "Retrace.""