I picked up a 390 12 gauge a few months ago, and just picked up a 390 20 gauge because I liked it so much.
I hate to tell you to go somewhere else, but go to Shotgunworld.com and go to the Beretta lovers section. There is a lot of informatoin there.
Among the recommendations i have read is to clean the piston with 0000 steel wool and not to oil it or clean with Hoppes as that will cause more gunk to gather there.
I remove the barrel and clean with a snake.
The bolt assembly is easy to pull out, although I found reassembly to be easier with shotgun vertical with the butt on my workbench.
the trigger assembly comes out once you tap out the pin. the first few times you may need to pull the trigger assembly forward, but then it supposedly is easier to pull apart.
I have not messed with the recoil spring or much more than that so far, but I've only put about 500 shells through mine so far.
Fun gun to shoot.
Not as many parts as the 391, but the videos are still instructional as to how to clean.
Once you clean and have the breach open, you will hear the piston go up and down as you move the gun. That's actually a sign of a nice clean gun, not anything to worry about.
Lots of info over at shotgunworld.