Getting comfortable with the core game is the most important thing. Honestly, the AT's that exist are fairly situational and don't create a gamebreaking knowledge gap like L-canceling and Wavedashing did in Melee. Learn the basics first. That's the most important thing, since Brawl is a different game entirely from Melee.
Know that Brawl's engine rewards defense over offense in most situations. Decreased hit-stun and shield-stun relative to Melee make aggressive styles less effective than they used to be. Combos are now shorter in length and DI has been buffed considerably. Smash DI also has a much larger time window than before. Recovery is not much of an issue, as almost every character can make it back to the edge without any trouble (tether recoveries are an exception). Though this is good for recoverers, edge-guarding is a little more difficult. Also, edge-hog frames have decreased by about half, so edge-hog is a lot harder to do now.
I feel that the way Brawl's engine works, the progression of a match is fairly predictable. Camping is now a viable strategy, and patience is a must for both players. In my experience, the match progression looks like this:
1) Both players defend using a camping strategy of some kind. Projectile spam is very popular for this step as it can pressure as well as defend. Characters without projectiles are at a disadvantage for this step, but can still annoy.
2) One of the players' defenses harasses so well that the other player is placed in danger or extreme inconvenience. The harassed player realizes that continuing to defend will cause him to lose.
3) The harassed player in danger is forced to approach. Approaches also require patience, but can be done effectively. Walk+PowerShield, Short-Hop+Air Dodge, and attack approaches all work well, but the emphasis should be on not getting hit. The closer the approacher gets to the defender, the more pressure is applied until the defense strategy's failure risk increases beyond a tolerable range.
4) Rock, Paper, Scissors. The approaching player is now close enough to hit the defending playerr. The defending player and the approaching player choose a strategy that they believe will result in a hit on their opponent. There are many different strategies to choose from, including shield-grab, poking with a long range attack, quick high-knock attack, hitting with a move that can combo, etc. This phase lasts as long as both players are near enough to each other to connect a hit, and "defender"/"approacher" role can change many times, even as often as after each hit.
5) The characters are separated by a high-knockback move. Edge-guarding may begin if a player has left the stage, in which case the recoverer is the "approacher" and the edge-guarder the "defender."
6) The process reverts to (1) or (3), depending on how much danger each player is in after 5.
As far as AT's go, read the list only once you know the basics and have established a playstyle. AT's in Brawl are mostly situational as the "win button" for Brawl hasn't been discovered yet (if it exists). Therefore, Brawl AT's should be used as supplements to your core strategy, not the center of it. Play with the ones relevant to your character and your interests and experiment. Some may be pure crap to you, while some may be just the thing you were looking for. Also, while some characters don't have "character-specific" AT's, that doesn't mean they automatically lose - Marth performs consistently well in tournaments even without a long list of personal glitches. Main who you like, not who has the most glitches.
Combos, though shorter than in Melee, are still useful. I don't know of a single fighting game that punishes you for taking a free hit, so use combos when you can.
Click here for the full list of combos in Brawl,
and here for the list of Chain Grabs. Like AT's, combos are supplemental tools, not cornerstones. None is so good that it will win you the match. All combos are proven useful, however, and can help you build damage faster than usual while minimizing risk.
Always check the character-specific boards for the stickied character guides. Know thy character. There is no better place than that character's "official" FAQs, as they can tell you important things you never knew about your main. You may find a use for a "useless" move or technique, or even fill a gap in your core strategy. The guides are written by talented researchers who invest a lot of time in their work, so take that knowledge and benefit from it.
Finally, remember to have fun. If you're not enjoying the game, why are you playing it? It's a whole different animal from Melee - many things have been simplified, a few more complicated. It's a very enjoyable game in its own right. Hope you have a great time with Brawl. See you around the boards.