Ok, understand that Advent capital ships are in a league of their own. They all work together and syngergize so much that Vasari and TEC caps just don't compare, although unfortunately it sounds like you were victim of an Illum spammer. Pure Illum spammers unfortunately aren't great players, they just spam the best ship in the game.
Capital ships are not meant to be the bulk of your forces, but rather to be a support ship for your fleet. TEC caps are rather....direct in their function.
To utilize caps properly, a good rule of thumb is that if you're at tier 2 population cap, you should have either tier 1 or tier 2 teched for capital ship crew slots, depending on how much you like caps. If you're at tier 3, then tier 2 or tier 3 capital ship slots.
Capital ships need a fleet to support them, and capital ships are truly great brutes that support the fleet.
I don't think they were outrunning ur cap fleet. All capital ships have speed of 600. The Akkan (colonizer) has a speed of 475, so yes, that one will never keep up. Illums have a speed of 500. That's the numbers there, though I don't know if they've been changed at any point.
In terms of each TEC cap, the Kol is meant to be a bastion, an absolute rock in the fleet. The same goes for the Marza to an extent. Obviously, this ship's for sieging, though the nuclear warhead is useful when aimed at clusters of enemies.
The Dunov is invaluable if you want a bunch of TEC caps. This cap can restore shields on ur caps. Every ability on this ship is a beaut. 2 of these can prolong the lifespan of your caps greatly.
The Sova a great damage dealer. It has itself, which is actually kinda weak for a cap, but then it has missile turret and squadrons of fighters/bombers. The damage output is actually quite great. Embargo is best used in rushes.
The Akkan has coloinize which will always help you expand much quicker early. Armistice, if you reach level 6, is an ability that keeps getting mixed reviews, but I still insist there's ways this ability kicks ass. The ion bolt is great early on. Speaking specifically in ur case, I'm guessing the Illums had a mothership accompanying them? ( It was either casting Malice or Shield Regen, kinda looks like a space slug)
If you aim the ion bolt at the enemy cap right as it fires whatever ability it's using, you can stop it with ion bolt. So lovely, and it brings an enemy advent fleet on par with you.