If it already has a scope and you want to sit and slow fire at targets for accuracy the trigger would be the next step. Optics and triggers are usually the first 2 upgrades. A RRA or Armalite 2 stage trigger would work fine and not break the bank, if you want to go all out a Geissele would get my vote.
I would clean it after every range trip, in the barrel a foaming cleaner is easiest, clean the foam out with several patches followed by a patch with light oil then a dry patch to remove the excess oil. A drop of oil on the trigger and hammer pins and on the carrier rails. If you shoot a lot of rounds(over 100) at a time you may want to carry a lube bottle or spray to the range, just keep the carrier rails wet, maybe shoot some through the 2 holes on the side of the carrier onto the bolt.
You don't really need to disassemble much. push the rear receiver pin over and break it down like a shotgun, pull the charge handle and the carrier out the rear. pull the cotter key on the side of the carrier then remove the firing pin, twist the square headed cam pin 90 degrees and pull it out past the carrier key, pull the bolt out the front. clean everything, light coat of oil(I use Mobil 1) and back together in reverse make sure the bolts extractor is facing the side of the carrier with the dish and ejection port on the rifle.
The bolt must be all the way forward when sliding it in the receiver.
You'll get 50 million opinions on how to clean and which oils to use and how often, just don't pour a can of oil down the barrel, excess oil in the barrel can cause high pressure when fired.
I'll look around for some good assembly and cleaning articles, if I find some I'll sticky them in this area.
hi perf barrels and bolts
The Nextgen of 308 rifles