Welcome to the game, and to the forums
From your post and the screenshot I'd suggest:
1) Take out 1 homeworld, then the other. If you work on them in parallel you'll face twice the counter-response until one's down. This is still workable, but is probably best avoided for one's first HW assault.
2) Warheads. Specifically Lightning Warheads and perhaps some Armored Warheads. You'll need to destroy any Core Warhead Interceptors on the planets you want to hit, and the warhead detonations themselves cost AIP, but there's pretty much nothing better at smashing a high concentration of AI forces when you
really need them dead.
3) Yea, the AI HWs hard-counter the Hive Golems. Perhaps I'll revisit that later as normally I try to avoid just shutting something down. But they were cleaning AI HWs by themselves at one point, etc. Anyway, I suggest reserving them for defense against any of the AI's counter attacks from the HW assault (if it runs that long). The response can be pretty brutal, but two loads of bees eat "brutal" for breakfast.
4) You've got a lot of hacking points left, so there's a bunch of things you could do with those to help. The most direct is probably to run a sabotage hacker on the AI HW while you're doing the main assault. If it gets to run long enough it will take out the forts and (iirc) the forcefields. Less-direct ways to use them would be to hack for knowledge on AI planets you don't plan to take, downloading designs from fabricators (I'm assuming you already have mkIV capacity for both fleet ships and starships, but if you don't then the hacker can help there). With those you can simply build a bigger fleet, which always helps.
5) Some of the AI HW core guard posts are nastier than others: on Murdoch you want that Grav Reactor dead, probably first; On Jelindor none are quite that nasty but the Implosion Drone and Shredder Drone ones should probably be taken down early.
Anyway, good hunting!
