Welcome to the forums, and to the game
Also, tip for next time. If you HAVE to take a planet next to a coreworld, unlock Warp Jammer command stations and place one there.
Yea, the warp jammer would have saved you a lot of pain with that core world. 5k knowledge isn't cheap but it's cheaper than a core world with multiple hours of reinforcements.
I'll also echo the "scout the rest of the map" point. Gotta know where everything is. Particularly at this stage make absolutely sure you've killed all the datacenters and (if you have the first expansion enabled, I haven't looked at the save) coprocessors. Doing that will reduce your AIP. If there's a super-terminal in the galaxy strongly consider hacking that for AIP reduction.
If there's no super-terminal, strongly consider hacking planets you know you won't take for knowledge. The AI's response to hacking can get pretty crazy, but if you haven't done any yet you should be able to steal a planet or two's worth of knowledge (or AIP reduction, if using the superterminal) without too much trouble.
Strategically speaking you want to avoid situations where you're facing a massive concentration of AI firepower. Tactically, since you're already facing that, try to fight it on your own turf. As the others have said: turrets! Generally speaking a cap of turrets has
three times the offensive punch of a mobile ship cap of the same mark. Also, the tactical usefulness of grav turrets is difficult to overstate. Spider turrets are also pretty brutal against stuff vulnerable to engine damage.
Speaking of that, give Riot Control Starships a try. The mkI versions are already available without spending knowledge and can wreak a lot of havoc on enemy engines. They can also mount tractor modules to drag a chunk of enemies to a planet or place more suitable for their destruction. The mkII riot starships can mount tazers, which are very brief aoe-paralyze that can substantially cut into enemy dps because paralysis suspends the reloading process. The mkIII riot starships can get some heavy-duty tractor modules for more theft, and some gravity-based tomfoolery.
You could also look into MkI Fortresses for defensive purposes. Extremely expensive in metal, crystal, and energy but if you've taken all that territory you should be able to support them. MkI forts pack an enormous punch. So do the mkII and mkIII, but there's a degree of diminishing returns knowledge-wise there.
On the carriers specifically, just popping them and dealing with the combined units may be best, since much of the above doesn't really apply to them.
I could go on about tactical options but I also need to get to work
In general I think you have a good scenario here for experimenting with various ways of beating large AI forces. Even if you don't win this game (I think you can, though it might require some cheese) that experience will be very valuable next time.
