Vasari are a pretty offensive race that really rely on exploiting their many unique advantages to the maximum. Stat wise per cost they aren't as effective as the TEC or Advent, so using them most efficiently requires more skill I think (certainly to the TEC).

Early on, you get the advantage of tier 1 LRF and phase missiles. Assailants and Sentinels will probably be your bread and butter early on. The Skinantra carrier is a great ship, use it to heal or spawn bomber to quickly take out the enemy cap. Late game they get many interesting advantages; phase gates and Kostura's for mobility, subverters and mine layers if you want some very effective micro heavy strategies, plus just the sheer offensive force of maxed out phase missiles.

You biggest problem sounds like its not faction specific though. Most newish players do not expand nearly fast enough, tend to do too much research early on, spend too much on static defenses. All of which slows their economy, development, and fleet build up down, which is certainly the case if you think the hard AI seems to swarm you quickly (unless its a very small map that shouldn't be happening).

  Speaking of late game comps, I usually have tier 1 Assailants and Sentinels early to mid game, but late game my friends start to transition into heavy combat cruisers, and massive amounts of fighters. Usually by then, I try to get carrier ships with bombers and mow things down, but strike craft get int the way. I guess I need the sentinels late game to stop the fighters, but while the flak are going to work, their HC's seem to be doing double to my assailants.

  Is there an ideal late game fleet mix the Vasari are supposed to strive for? If there isn't, is there a philosophy for a core backbone? Or should I cut down my late game ship production and build superweapons?

  And speaking of super weapons, are the Vasari phase gates worth it besides the mobility? I like them alot, but I'm not sure if the fleet you get for your money is a bargain. I used to build these, but got annoyed with the amount of military labs I needed to build, and simply stuck with building an Army, with about 3-5ish military labs endgame. 

To clarify, I play on smaller maps, where I end up with 5-7 planets, and as far as super weapons go is it better to plan to make it, or to use the resources and time to make a giant fleet instead?

Early on, get phase missiles and assailants.  When available, get Charged Missiles.

And yes, you should.  I typically bring cruisers in later in the game, but light carriers are always useful to have a few around because of how powerful SC are.  The Ruiner is also a vicious little thing if you use it correctly.  Don't use it to reinforce your gravity wells.  Wait until a significant portion of the enemy fleet is disabled and then lay mines in the middle of them with it.  Works wonders.

I could go on, but I have class, so I'll try to give a more complete explanation later.

Shield bypass is the best weapon bonus period because it also bypasses shield mitigation. Normally it reduces the damage done by 15-57%, but when phase missile successfully bypass shields it does full damage directly to the hull.

You should use mine layers with subverters to disable a lot of ships and deploy mines around them. It takes a lot of practice to use well as a few undisabled flak frigates can shoot the mines before they can explode, but its a way to inflict a ton of damage very quickly. Its not a tactic I would recommend for new or casual players though.

I generally go for flak first because they have a good amount of health and counter the counters to Assailants.