Noobs giving noobs advice.
Vasari econ not good? Slow starters? Uh, did their assailants get put as a Tier 4 tech while I wasn't looking or something, or are they still Tier 1? Sounds like they're pretty fast out the gate myself, especially since they're widely known as the "rushing civ".
Do yourself a favor and disregard most of the above posts about vasari. Am I being cocky? Yes. Am I going to give you better advice here in a minute than about 95% of whats above? Yes.
First off, the two very combat worthy (LOL) ships that Vas fields are the assailant and the enforcer. If your opponent is far enough away and/or you can get a feed from your teammates, I would typically recommend teching to enforcer as quick as possible and making a ton of those to start with, adding assailants after you have a number of enforcs (about 12-15). Enforcers counter Illuminators, the most spammed unit on ICO. They also provide a meatshield that absorbs damage while the assailants, which do awesome damage but are relatively frail, pick away at your enemies.
If your opponent is close, then tech assailants right away and try to put assailants at his worlds before he starts spamming his illuminators. Target the frigate factory first, and then the Hostilities. Usually you dump everything you have plus a little your buds have (AKA feeding) into assailant production so you can overwhelm him early.
Now med distance away is where you're sorta fucked as Vasari. The advent player will be able to tech illums before you get any functional number of enforcers out, and his illums will eat your assailants in 75% of the straight up battles. My advice? Make sure you have every neutral that you can get, expand as much as you can behind your home world so that he can't easily access your breadbasket, and build repair bays/turrets on your homeworld while spamming the crap out of assailants. Once you have a large blob of assailants, start teching up phase missiles, health upgrades, and Armor upgrades. Why? Because a 6 Armor assailant with as much HP as an enforcer and phase missiles at 20% or 30% will eat illums for breakfast, burp, and ask for more.
As far as vas combat strategy, the piper can't give away all the tricks. I doubt half of you guys care enough to do the stuff I do anyways, since it's very micro intensive and the payoff is relatively small in the grand scale of things. I will suggest that you leave your Evacuator, your first vasari cap 100% of the time, away from the battle unless you're taking down an enemy mothership. Why? Because you want the damned thing to get experience so that you can get drain planet so you have to leave it near the combat, but you don't want to lose it in 20 seconds, which is what seems to happen every time the egg is mixed with assailants right on the front line.