Welcome to ShootersForum, Izzy...rules are simple, be nice and join in!
You'll find that most of us on here love our bolt-action .22LR's because they are what we learned to shoot with, they are what we still do a lot of practice with and they're just plain useful and fun!
But, they're not sufficient for deer, so that's what you're asking about. For a new guy looking for a big game rifle, I think it would be hard to go wrong with one chambered for 270 Winchester or 260 Remington. As for the action, if I were in your shoes, I would go to the local pawn shops and try to find a used Browning A-bolt, Savage 110, Winchester Model 70, Remington Model 700, or a Ruger M77...all bolt guns and all likely to give you everything you're looking for in a rifle.
Now, if you've got a fair-sized bankroll and would like to do something a little "different" pick up a T/C Encore in 260 Remington and then buy a .22LR barrel for it. You'll have a very accurate gun that will do both of the things you originally listed and if you find one used, probably be into it for less than $750 (plus two scopes). Also, if you really get bit by the huntin' bug and wind up going to new places and shooting bigger critters, you can always buy a 300 Win Mag barrel for it...lots of flexibility in the Encore.