A lot of it depends on the kind of map you're playing.

Generally speaking, most people will recommend that your first capital always be a colonizer for purposes of expanding faster. And in most cases, this is also good advice.

It's really hard for an Advent player to justify starting with any other capital. Their colonizer reduces upgrade costs on the new colony by 20/40/60% and at the same time its AE shield renegeration buff is great for the fleet. The Vasari one is next in line. Its colonize bonus isn't as great (build speed reduction), but it's a pretty efficient capital ship killer in the early game due to its DoT ability that bypasses shields and damages hull directly, and in longer games it can stop a group of ships from phase jumping and suck enemy colonies dry for money and resources. So, it's a great all-around ship. The TEC probably suffer the most with their colonizer. Its bonus is to give you free resource extractors on new colonies, but in combat it's just not a very impressive ship. It has a passive aura to increase range of your fleet, a short-duration Ion Bolt that interrupts everything and stops the target ship, and its ultimate is really only used to retreat (prevents all combat in the grav well).

The smaller the map gets, though, the more reason you have not to start with a colony capital (especially TEC ). Colony frigates now start with full antimatter when you build them, so there's no delay on colonizing anymore. You also won't need to colonize nearly as many planets, so the various bonuses don't play as big a role in the grand scheme of things. And, at the same time, you generally have smaller fleets so combat-focused capitals can shine a bit more - especially bombardment ships since those colony capitals will take ages to nuke down a colony unless you build a bunch of siege frigates to help out.

For the other ships, it really depends on your style of play.

The Advent is probably the easiest to decide on because their capitals give a good flow of abilities. For example, their battleship can taunt nearby enemy ships and make them all attack it, their battlecruiser at the same time can put a buff on the battleship that returns a percentage of damage taken to the attackers (that have no choice but to attack the battleship after it taunts), and the mothership regenerates its shields so it can keep the chain going. Just one example of what you can do with a mix.

The Vasari and TEC don't have quite as developed capital ship synergy, so it's easier to mix and match.

The Vasari battleship is a pretty monster damage dealer with its weapon cooldown reduction ability and its ultimate infects enemy ships with volatile nanites and makes exploding ships damage their nearby buddies. The Vulkoras desolator is great early on against frigates with its high damage phase missile swarm ability, and it's an extremely good planet-killer since it can create bombardment platforms and its own bombing cooldown decreases (and it also does extra damage against buildings). The carrier is probably the least interesting, but it has a hull repair cloud ability that can help keep your fleet patched up.

For the TEC, it's rare to find a fleet without at least a Dunov, since all of its abilities are very useful. It recharges a massive amount of shields to a single target, which is great for keeping up a friendly capital. It has an AoE Antimatter drain that also deals a little damage, great for breaking the Advent fleet setups that rely very heavily on antimatter. And it has an ability that makes nearby fighters smash into the hull of the target ship, dealing damage to it and destroying many fighters in the process. The Marza, on the other hand, is both a monster ship killer with its ultimate, and a pretty damn good planet killer with its Raze planet ability. It's also a very high damage output ship in general, so you can't really go wrong with it. The AI isn't that great at interrupting its ultimate ability channel, so you'll probably find it makes things a bit too easy. The Kol is more of a defensive ship with its deflection shields to reduce incoming damage. It does have a gauss cannon ability that's great for buildings (ships start mitigating a lot of damage from it) and its ultimate makes it a temporary beast on the battlefield. It's also got fairly decent anti-fighter flak cannon ability to round it off.

Well, hope this helps a bit.

Providing a rundown on ALL the capitals would be rather lengthly for a post, but their details can be found in one of the four stickied threads in the Strategy section (link to a Google Spreadsheet).

Another common combination of Advent capital abilities is the Radiance's Cleansing Brilliance with the Progenitor's Malice. For 8 seconds, CB will do 250 damage/sec to targets in a cone in front of it, while Malice will calculate the total damage taken by affected ships over a 10 second time and deal an additional 30% of the accumulated damage to those ships when the effect ends. The Advent capital carrier, the Halcyon, is more focused on supporting its strike craft, combating enemy strike craft, and supporting the surrounding fleet. Version 1.1's new focus on strike craft makes the Halcyon slightly more valuable, since its Telekinetic Push can deal physical damage to enemy strike craft and slow them down for a considerable amount of time. One of its passive abilities, Amplify Energy Aura, decreases the cooldown times of all friendly Advent ships by 8%/15%/22%, which made it popular in tandem with Illuminators. The Revelation, on the other hand, is probably used mostly for its Reverie ability, since it can disable enemy ships (including capital ships) until the effect expires or the ship takes a certain amount of damage. Its Guidance ability, though it decreases ability cooldown times (by providing a 40% increase to cooldown rate), is a rather "meh" ability and Clairvoyance simply gives you viewing of any known gravity well (does not matter if a ship went there, you just need to know a phase lane leads to it). In contrast, Provoke Hysteria is probably one of the faster ways to take out developed colonies (i.e. planets with maxed Emergency Facilities) since it damages 1% of planet health per second and kills 1% of the total population per second for 40 seconds. It's also rather freaky, since you end up hearing the screams of the population being murdered...

and if you combine the two, you get the Vasari's big intsa-fleetkill combo.  it works best with 2 Devastators and 3-4 Desolators, if ur against a large fleet.

SageWon - that would ge GREAT! A summary of the CAP abilities, that they are useful for, some hints on how to pair them, and suggestions for beginners on what to consider would be VERY helpful!

We can look up the CAPs anywhere, but what is missing is what they do. For instance, what abilities should be micromanaged? Are some of much greater worth against some enemies than others? What are some suggested combos?

I'm just learning to manage the CAP abilities instead of letting the go on auto (Marza barrage is SO much fun!!), so hints are more than welcome...

HydroAC

It is about 20 pages. 

And doesn't include some of these.  HydroAC, I can point you to a few posts that might help.

Including these particular posts by: Raging Amish re: TEC, jonathan421595 re: Advent, and Raging Amish again re: Vasari.  And one on Cap Ship build order.  And don't forget these by Mongoose22 and Zyrxil.  But remember, some of these are old posts, and things have changed, due to 1.1 fixes/changes (esp. Missile Barrage).

big one that needs to be micro'd for Vasari: Phase out hull, on the Antorak marauder.  that's probably the one that should be micro'd, or turned off.  never use it on autocast, because like that it's worse than useless.  it will target the ship you are attacking, forcing your ships to switch targets, and it is a pain to retask them each time.