For Advent, it is almost always Prog...you see people try the other caps but if they do well, it is usually because they outplay their opponent, not because their cap choice was a good one....the prog helps win fleet battles with malice, is good for suicide and eco because of the colonize, and helps against missile barrage and Orky rushing with shield regeneration...there is no other cap that does all of those things...
If I had to pick a second cap for Advent, I'd say Halcyon...it has a decent AoE and the SC give you an edge against Orky rushing and early game battles...there really is no reason to start with any of the other caps as the prog or halcyon can do what they do, but better...there is one exception, and that is if somehow you know you will be facing a Corsev rush...in that case, a radiance probably would be better, but I'm not sure how you'd know for certain that was going to happen...
For Vasari, it is almost always the Egg...the colonize on it helps Orky rushing better than any other cap since colonizing the planet is of immense importance...nano-disassemblers also helps fight Orky rushing and drive off annoying capital ships...the other two abilities are very good at all stages of the game, so it is an excellent long term investment...
If I had to pick a second cap for Vasari, I'd say Kortul...it can take a beating early game and limits the ability usage of enemy capital ships...but really, this ship is a far 2nd from leading with the Egg, as the kortul really only shines mid and late game when titans and SC spam come on the field...
For TEC, you have a lot of options...in fact, you can successfully lead with pretty much any ship other than the Kol...personally, I like the Dunov the best...EMP does AoE damage to shields and limits the enemy's ability to use their cap abilities...it counters the fleet-killing abilities of the marza, corsev, and prog, it can interrupt and counter missile barrage, and it does all this while providing an AoE of its own...the really only downside is that it does nothing to help against Orky rushing, so you will be relying solely on sheer frigate power for that....