Any of the 150, 174 and 180 gr rounds should do just fine. I would lean toward the heavier weights for bear, however. On a personal note, while I've had good luck with either Winchester or Remington ammo, I generally shoot the brown box PMP 174 gr. sp. ammo. It's usually much less expensive and just as accurate. I find it at Dunhams and at gun shows, although it has been a bit scarce of late.
Unless your .303 is one of the newer commercial models, its always wise to have it checked out by a competent gunsmith.
The .303 is a great gun and cartridge. Good luck and enjoy!