Hello, welcome back.
One of the features that may be new to you is the addition of controls presets. These were made during the controls code overhaul project as a way to preserve controls bindings between pilot selections and also a handy way to share presets among others.
There may be a preset floating around that has a sensible binding scheme for gamepads.
I used to use a somewhat straightforward preset, which I don't currently have easy access to due to working so much these days.. but that's another story. Anyway, the scheme is so:
D-pad is for energy management, left, up, and right are for weapons, shields, engines, and down is equalize.
ABXY buttons (or equivalent playstation geometry) are targeting. I don't quite remember which one was which, but it was something to the effect of nearest hostile, boresight, targets target, and maybe nearest bomb.
Shoulder buttons were cycling weapon modes, the trigger or second shoulder buttons are for firing said weapon.
Then finally secondary fire rates were bound to the "start" or "select" buttons, afterburner was a joystick button press, left stick was pitch and bank, right stick was throttle and roll.
...if I can find ample time and (remember) I'll toss a preset file your way.
P.S.
Oh, if the F1 help overlay wasn't helpful enough, there may be a FAQ on the freespace wiki that goes over everything in detail (just put FAQ in the search bar)