A good, cheap rifle for deer, IMHO, is a Marlin 30/30. That was the first rifle I ever bought. With a decent 3-9x40 scope (which will cost as much as the gun), it is about as deadly as anything. Its easy to carry, ammo is cheap, and recoil is moderate. It won't make any spectacular long range kills, but when used within its limits, it is very effective.

If your looking for a longer range option a 25-06 or .270 would be up to the task. I have a 25-06, and have shot .270's. Most any brand of bolt action rifle will provide you with satisfactory accuracy as long as it is floated correctly. I haven't ever shot any of the single shot rifles, so I can't comment on them. Just keep in mind that topping your rifle off with a good scope will cost $$$.