I play a lot of vasari lately, mostly because I believe they actually stand a chance against late game advent unlike TEC. Generally my play style is as follows.
Immediately build a cap yard, crystal mine, scout and upgrade your population. I always choose the desolator for it's early game AOE (it really really chews up lrm's) and later on because a cap ship generally turns into a siege weapon. (siege platforms is more effective then the passive siege bonus).
Get a colony ship, two metal mines and then another scout. You will need to buy more crystal to grab the colony ship. Use your scouts to search for neutral resources. If the resources are guarded use way points to circle the grav well while getting ant-matter. Sell excess metal/crystal for credits (you will always be short on credits if you get even a few neutral resources).
As soon as your cap ship is made send it to the nearby asteroid and begin clearing. Upgrade population ASAP (rule of thumb is to always do this).
While you are clearing you have time and money to build a single weapons lab and research assailants.
If you have another asteroid close by clear that. If not I get 3 assailants and clear out the nearest terran/desert planet. Additional assailants are constantly purchased and sent to my cap ship. Always send the cap ship in first and wait for him to get aggro BEFORE you send in assailants. Militia kodiaks will attack your assailants so kill them first. If an assailant is taking damage from two or more kodiaks retreat it. After the kodiaks are dead kill LRM's then cobalts then siege then flak. Practice this and you can take a desert planet long before someone who did civi research can get their first ice planet. Research repair bays at some point. If you are going to get hit with pirates figure out where they will go and build one repair bay. If you are busy clearing a planet or fighting an opponent two or so missile bays by the repair bay can take care of early pirate attacks. If you want to spend time fighting the pirates micro your assailants around inside the repair area so that you don't lose any.
Build up assailants to the first fleet cap, and then upgrade to the next tier. You want to constantly be pumping out assailants at this stage. Furthermore, excess cash you end up with (from selling metal which vasari usually have tons of) should be used to build more weapons labs. You need to get to five of them ASAP before doing any other research. Hold off on building assailants when you need resources to upgrade planet populations.
Once you have 3-2 places near your homeworld colonized and built up look for a forward base (desert/roid/terran) close to your nearest opponent. Take it before you opponent can and build 2 frigate factories and a repair bay (before you even build extractors). By this time you should have a level 4 cap (2 in phase missile, 2 in siege platform) and 14+ assailants (depends on how fast you can clear and colonize). You should also hopefully have 5 weapons labs. Research enforcers, the first three hull upgrades, and both reintegrate upgrades. While you are doing that research begin your attack. Focus on killing your enemies frigates with your assailants. If they focus fire on your cap ship fly it in and out of their fleet just often enough to hit your phase missile swarm skill. Retreat it back to your repair bay if it starts taking hull damage. While the attack AND the research is going on start building enforcers at your forward base, but don't send them into the fight and give away your hand. Once you have 5 or so assailants send them in. While fighting at enemy worlds make sure to hit construction drones, ESPECIALLY if they try to build static defenses. Take out repair buildings first. If they have a bunch of turrets then skip that planet and hit another one. If you find a planet with a lot of military labs make sure to take one or two out, especially if you hit the world when your opponent is somewhere else. Lack of military labs means they can't counter you enforcers easily.
Once your enforcers enter the battle (5 or 6) you have two options. If they have carriers for some reason research sentinels and get the AOE missile upgrade. MAKE SURE TO TURN OFF THE AUTOCAST ON THIS ABILITY OR YOU WILL LOSE A LOT OF UNITS FOR NO REASON. Manually turn it on when your sentinels are in position to defend your cruisers (as an added bonus any surviving assailants now doe AOE damage, but manually cast it because like sentinels the autocast on this ability gets them killed).
If your opponent has no carriers then invest in subvertor research, get a sixth lab, and research the disabling ability. Manually chain it on groups of enemies for an easy victory.
If you played correctly you will overwhelm ANYONE who tried to do any kind of civi research, especially in a 2v2 or a 3v3. At the same time your economy will be fine thanks to selling neutral resources and the fact that you've colonized terran/desert planets.
Things to fear as vasari are repulsion, and well repulsion. Actually, that's pretty much all you have to fear. Repulsion will mean that you must use assailants to take out guardians (tough to do) before your enforcers can do anything. Unfortunately assailants are far inferior to illuminators. If you have subvertors by the time your enemy get's repulsion try to disable anything you can to give your assailants time to whitle down his guardians. TBH at this point in time I haven't played a GOOD advent player using repulsion, although I have abused the strategy many times myself so I'm not sure if that will work at all. I've beaten POOR advent players who read about repulsion and must have thought it sounded pretty fun, but I don't consider that to be a good way to learn counters.
Another good strategy is to rush up to returning armada with a TEC player feeding you cash. Build a few Phase gates(connecting important worlds) before doing all the research, mostly to give your phase gates time to build up anti matter (something they are often short of). Build a phase gate at each planet for travel purposes. Then at your most defensible planet build multiple phase gates (planets can hold 3 and roids can hold 2). Skirmishers from RA can be scuttled, as well as transports. Upgrade your fleet cap like crazy, and research the scuttle tech. Spare cash should also be used to build 6 weapon labs so that you can supplement your free fleets with subvertors. RA backed up by subvertors is basically a win against anything except maybe a strong advent fleet consisting of repulsing guardians, defense vessels, masses of illum's, a level 6 radiance and a level 3+ mothership. That would be a fun fight actually.