Pat and Ian have covered the three-color angle pretty thoroughly, so this
answer focuses on aggro-vs.-control.
Aggro-vs.-control matchups tend to revolve around board presence. I
look at it this way:
The aggro deck is trying to put a lot of threats on the board, and the
control deck is trying to remove them or build his own board. Every
turn that the aggro player has a stronger board presence brings her
closer to victory, because she gets additional incremental value from
each of her threats.
For example, any turn that it's on the field and your opponent doesn't
have a blocker, Goblin Guide gets to do 2 more damage.
At the beginning of the game, the control player's immediate priority is to
"stabilize", usually by getting to a sweeper like Day of Judgment.
We often talk about the "card advantage" build into a many-for-one spell like
that, but the momentum swing it provides is usually more decisive.
I think your priority as the aggro deck isn't so much avoiding card
disadvantage as it is maintaining pressure, through a mix of two strategies:
"Be fast": play an all-in game where you dump your hand out as fast
as possible and try to get your opponent to 0 life (or 10 poison)
before he has a chance to play many cards. During deckbuilding, this
means prioritizing low-curve, cost-efficient threats like Goblin
Guide and Wild Nacatl, and packing in "guaranteed" damage
like Lightning Bolt to get in that final blow after you've lost
control of the field.
"Be tough": develop a respectable board position to create
pressure, then maintain it despite removal and disruption; you're
still playing aggressively, because your opponent's "endgame" probably
trumps yours, but the idea here is to deny your opponent the ability to
stabilize early and easily. During deckbuilding, this involve persistent
threats like Doomed Traveler, Chandra's Phoenix, and
Thrun the Last Troll, or alternate lines of attack like
Shrine of Burning Rage.
Note that these aren't so much competing gameplans as two facets of
the same goal: getting your opponent to 0 life quickly and consistently.
A good aggro player will
tailor her current approach to the strengths and weaknesses of her
opponent's deck, so it's valuable to build decks that can play both "fast"
and "tough" (and are, ideally, at least a bit of both at the same time).
If your deck is running blue, you also have recourse to countermagic
-- which you'll often see used in tournament "Bant" (green-white-blue)
and "RUG" (red-blue-green) list. This is the basis for the
"aggro-control" style of deck, which uses control cards to protect its
board position. Forcing your opponent to wait until he has enough mana
to play Day of Judgment with counterspell backup is often
backbreaking against straight-up control decks. Generally, though,
avoid running specific answer cards unless they are also threats, so that they don't
just languish uselessly in your hand while you wait for the right
moment; e.g. Qasali Pridemage is a better aggro choice than
Disenchant because a 2/2 exalted is always useful even if you
don't have an enchantment or artifact that you need to blow up.