I've seen threads like these before, but nobody evolved into a comprehensive strategy guide for Vasari. If you know of someone who has actually done this, I'd appreciate a link. So, here we go with a collection of tips to help us noobs make Vasari great...
In general, Vasari have the strongest and most expensive frigates and cruisers. In the military side, the phase missiles are their strongest side. However their economy isn't as strong as TEC's and the fleet doesn't have nearly the same synergy as Advent's. Vasari's greatest advantage is their advanced phase technology, allowing them to fight their enemy in multiple locations at once. They also have a few other complementary abilities but these are minor in comparison.
Early game
Build a number of scouts early. These are important for taking extractors in non-colonizable grav wells. Later in the game you could use them for hit-and-run strikes, eg. taking out enemy constructors (do they have Light armor?). Note that you shouldn't do that in early game yet, because scout rushing is a cheap tactic and will get you shunned in multiplayer.
Assailants have the greatest range of any unit and are a very useful damage dealer even in the late game, so build many for your first fleet.
Phase inhibitors can be built with just 2 mil labs so use these to discourage the earliest serious attacks, and to protect central locations in your empire.
Mid game
This is the part of the game where Vasari seems the weakest. Your best bet may be to tech up and pray that you don't get squashed by the other factions.
In the economy side, your strength is refinery technology, which is not that stellar. Research the harvesting upgrades, they're cheap for you. Trade ports are probably worth the sacrifice, depending on circumstance. If you can do raiding, you should make sure to make use of your Evacuator's planet draining ability.
For military, it appears you don't have any particular advantages. Just tech up and build a complete fleet like everyone else. Max out phase missiles. I've never heard anybody praise Vasari for their superior support cruisers, but I guess they're eventually worth it. Vasari's strenghts are raw damage and tanking abilities. You also get a dps bonus in friendly culture.
The only cap ship that is very good is the Evacuator space egg. None of the others is very bad as such, but there is little synergy between the cap ships. For this reason I'd think it's best to divide your forces into different fleets, each with its own focus and a cap ship suitable for the purpose. Use this for raids and defense in multiple locations at once.
Check a capital ship guide for details, but here's the quick runup: Devastator is generally useful, surround it with assailants and you can use it to protect those from enemy fighters. Desolator is good for taking out planets. Marauder will allow you to do hit and run strikes, eg. against the enemy hw, with its phase gating abilities. The Skirantra carrier isn't especially popular, but some people swear by it because of its complementary abilities. Used in pairs, the repair cloud can be very effective and of course it stacks with your repair bays. Using the replicate forces ability on friendly destructor frigates is a good way to devastate planets quickly.
Nano weapons jammers are your best defensive structure, so get those up.
Late game
Getting your phase stabilizer network up will mean lots of trouble for the enemy and is the whole point of playing Vasari. You can hit hard - thanks to your good dps and offensive abilities - anywhere you want, at any time.
Returning armada will allow you to get reinforcements at a discount, thus you can offset your economic disadvantages.
