The basics still apply: Progenitor first as Advent, scout voraciously, and spend your money prudently.

Beyond that, here are some mistakes I notice of newbies often:

1) Run your capital ships away early!  Players are much better about focus firing a capital ship than an AI is, so it will die sooner than you're used to.  Chances are if you haven't already started retreating a capital before you hear "our capital ship has sustained critical damage" it's already too late to save it (especially against Vasari).  Remember that retreating doesn't necessarily mean leaving the gravity well, just getting away out of the enemy's line of fire.  If you can do that without having to phase jump, power to you, if you can't, just get the hell outta there!

2) Static defenses are overrated.  The AI likes to run head-first into turrets, starbases, and all those other lovely (and immobile) defenders.  A player will go around them.  Make sure when you place defenses like these that they are guarding something, otherwise they may never get a single shot off.

3) Beware unconventional units like scouts and flaks.  In large quantities, these can give you a lot of trouble with conventional units like carriers and LRF.  Diversify your army, pay attention to what your opponent is building, and build accordingly.

4) Don't rush into trade ports, especially if your empire is small.  Trade is great... when a game goes on for a few hours.  A smart player won't let you survive that long, even if a more passive AI will.  Remember that capturing new planets is always a better investment, and trade ports require a fairly large empire to be effective.  I make sure I can build a chain of at least 4 before I commit to them.