Battleship - Devastator
Power surge - No idea how to use this. Advice would be appriciated.
It buffs the recharge rate of the Devastator's weapons (greatly improving it's damage output) and its shield regeneration. It's a pretty awesome buff.
Jam Weapons - Disables enemy strike craft in a small radius correct? High or low priority?
Depends on how many strike craft you're running into. Handy but not that great, in my opinion, as it doesn't actually damage the targets.
Disruptive strikes - Also, have no idea how to use this or what it really does.
Does exactly what it says - gives a chance to reduce the target's antimatter and increase the recharge of their abilities every time the Dev hits with its energy weapons.
Passive, and can be surprisingly useful against enemy Caps.
Volitile Nanites - This one is a clear winner. Saw the vid of the fleet of LRM getting wiped out and I was sold. Part of the reason I'm trying to learn this race.
Yeah, it's pretty cool.

Carrier - Skirantra
Repair cloud - Seems simple enough. Does this stack however? Should carriers be on the front lines?
It doesn't stack with another repair cloud, but it does stack with different repair abilities (like the self-repair on upgraded light frigates and heavy cruisers).
If you do have two or more Carrier Caps, just don't fire it from them all at the same time.

All Caps are pretty durable, so you can have them backing up the front lines. If this makes you nervous, Repair Cloud is still useful after a battle is over, to get the fleet back to prime condition much more quickly than through auto-repair alone.
The Vasari Caps have a lot abilities that make them well-suited to raiding well behind enemy lines, and this is one of them.
Scramble bombers - This creates one extra squad of bombers correct?
Yup. I believe the duration is twice the recharge, so until you run out of antimatter, you can actually get two extra squadrons of bombers out by firing it every time it recharges.
Microphasing Aura - Still a little foggy on this one. Is it useful? How do you apply it? Is this low priority?
It gives strike craft in the area a chance to teleport towards their target every few seconds, allowing them to close range faster (and thus increase their damage output). Also, for the second they're teleporting, they can't be attacked.
If you're not afraid to bring you carrier cap up to the front lines, where your strike craft are fighting, it can be quite handy (it's passive, so it's on all the time). If you're going to sit back from the action, though, it's less useful.
Replicate forces - Can someone explain this one and its tactical application?
It creates copies of the target friendly ship - basically it allows you to create new ships for 'free' (no cost in cash or resources).
Another of the abilities that allow the Vasari Caps to work well in enemy territory - if your fleet takes losses, you can replace them 'in the field', given time (it's antimatter cost and recharge are fairly hefty).
Colony Cap - Evacuator
Colonize - Vasari Colonize only increases build time correct?
Yup, colonies it creates will build structures quicker for a time.
Gravity Warhead - Slows down Enemy ships and prevents phase jumping. Is this an AoE or single target? How long does it disable phase jumping?
The duration is listed on the info card for the ability - I don't remember it off-hand.
I'm not sure if it's AoE or not - I've only ever used it to stop enemy Cap ships from jumping away, so I've not paid attention to whether it also affects nearby ships.
Nano-Dissasembler - This is a DoT? Appears to be the only dmg ability this ship has. Any advice on how to use it? Is this a priority over the other 2 abilities?
It's a nice DoT
and a hefty armour debuff. Very nice offensive ability for a colony Cap ship, in my opinion.
Drain Planet - Haven't used this one. Any advice would be appriciated.
Steals resources from enemy planets at a pretty high rate - even if you don't have time to kill an enemy planet before the player brings it back up, it can be very useful in 'raid' attacks.
Support cap - Maraurder
Phase out hull - Still haven't quite figured out how to use this. I know it takes one ship out of the fight, but how is it best to apply this power? Once the target goes out, it ceases to be a target, so the your Mararder seeks to find a new one, often going past the target ship. Do you have to manually sit there the entire time phasing it out and then stopping it from moving?
It takes a target enemy ship out of the fight for a short duration. Great for use against enemy Cap ships.
Distort Gravity - Speeds up your ships. Cool ability. Still need to master getting everyone in the radius.
This is a pretty awesome ability that can be used offensively (to close on enemy vessels, especially fleeing ones), defensively (to escape a battle) or just to get from one system to another quicker.
Subversion - Still trying to see the value in this one.
It's a 'raid' ability - useful when trying to harry the enemy rather than actually attacking in force.
If you don't use those kind of strategies, it's not much use, I agree.

Stabalize Phase space - This also is a great ability. This ship...ability wise, seems the best in the fleet.
Yup, this is another awesome ability, and particularly well suited to working deep in enemy territory.
Dreadnaught - Desolator
Phase Missle Swarm - Direct Damage, seems to be best ability.
Yup - hits more than one target, so best used on groups of enemy ships.
Deploy Siege Platform - Don't see the use of this one.
Helps you take down enemy planets quicker, it's as simple as that.

Assult Spec - I put points here just because they have no where else to go.
Does two things - helps you take down enemy planets quicker and does pretty massive extra damage to orbital structures. The second makes it handy in raids.
Disintegration - Haven't had the chance to use this one. Any advice here?
Massive DoT, and repairs the Desolator's hull at the same time. A great anti-Cap ship ability.
As I've mentioned, Vasari Caps have many abilities that allow them to survive deep in enemy territory and make them well-suited to raids on enemy planets, where the goal is to quickly cripple enemy planets and jump out before a defending fleet arrives.
At the same time, the solid damage abilities allow them to perform well in 'stand-up' fights.