AI Cores are mainly combat loot, so getting ready for combat (via pirate outpost) ist the first big step. Brigands (Lvl2 mobs) start drop cores with 15% probability, Lvl3 mobs and missions with higher prob.
You can perfectly go lvl 1 or 2 bounty hunting without any ai cores if you have your laser and shield equipped with all the other subunits that can be purchased at the pirate base and iltari (if you have your license) or found in space events.
Once you have Vee join your crew (around day 40) and the swarm comes for you you get tons of cores :)
There's also a walkthrough in the OP with more detailed combat help.
Glad you enjoy the game so far!