I agree with JDaveG, the Remington 700 is an OUTSTANDING bolt-action rifle, you can find them all over for less than your price point leaving coin for quality optics.There are other really good bolt-actions like Tikka and Sako, the Ruger precision is a really nice one as well, the Remington 700 is a tried and true (our military snipers use it) rifle and I would trust it for hunting, sniping, competition, et al.
If semi-auto is your thing, (it is mine) a battle rifle is simply a must, an M1A or FAL would be the ticket. You are not going to get one for under $1000 but might find a decent deal used. Bolt actions are going to be more precise, but semi autos will let you obtain mutiple hits on target in quick succession without sacrificing too much accuracy.