A BLR, perhaps?.308 or .30-06, you'll have the accuracy and range of a bolt gun, with the speed of a levergun.
However, they're not high capacity guns, and magazines are expensive.
A lever is easier to "learn" to shoot fast. You'll have to practice to be able to shoot a bolt action fast and accuratly.
A Lee Enfield is pretty fast to shoot for a bolt, but .303 is british, so it'll be harder to find than American calibers.
Personally, I'd either go for a Remington 700 or a Ruger. Have decent iron sights put on, a 10 round magazine and a good scope and you'll be ready for anything. Of course, some will tell you back up irons are useless, but it's better to have and not need than need and not have.