You are going to get many opinions as to which brand and caliber (Ford vs. Chevy, etc.) with this question, so take the responses with a grain of salt.
For around $500 (maybe less) you can often find a new Remington 700 ADL or a Remington 700 SPS. If you want a good quality rifle that is about the best you can buy for that budget. There are a lot of other less expensive choices but in the long run you will be happier buying quality. The ADL and SPS models are the same as any other 700 except for a cheaper stock. With a Model 700 you can upgrade it later (stock etc.) if you wish.
There is a large market in very good used rifles. You can get a lot more bang for your buck by buying quality used rifles (but not used scopes). For less than $500 you can find a Remington 700, Ruger 77, etc.
Buy the rifle and scope separately if you can. Package deals sometimes do not work out in the long run. A good scope with rings and bases will cost you a bit, but that is another issue.
For me it would be a .30-06 as a first choice and a .270 as the second choice. You can find ammo easily for these two cartridges and they are flexible enough to allow you to hunt larger game than whitetail deer if you ever decide to do so.