Don't confuse a good 'all-round' rifle on the one hand, with a best initial deer hunting rifle on the other hand.
For a first rifle, you want something that is very consistent/precise ('accurate'), relatively inexpensive when new, powerful enough to be deer-effective at moderate ranges, recoil-comfortable enough to shoot all day several days a week, and have relatively cheap ammo.
That would be a 308 or one of the 6.5 cartridges or 7mm-08. Maybe in a heavier-barreled Savage, maybe one of these:
http://www.savagearms.com/firearms/model/12FVSS
http://www.savagearms.com/firearms/model/12bvss
http://www.savagearms.com/firearms/model/12LRP
Potentially get an inch or two lopped off the muzzle of those if necessary to improve swing or balance, and increase bbl stiffness further. (Don't need any more than 24 inches for 308 loads). The rifle sort of has to feel OK, then get it properly fitted for length of pull, and then cheek height after fitting the scope to it.
You want to shoot the heck out of it in the first few years of experience, in addition to snagging your first whitetail. Develop good shooting skills, body position, trigger & breathing control. (Not just bench-banging or 'sighting in' a couple of days before hunting). Good precision is very encouraging. And you want to know that if you made a bad shot it was for sure your fault and not the rifle somehow, so you can trust the feedback. Keep pushing the distance envelope while keeping small groups, especially when prone or sitting.
You also want to be able to take long strings of shots before the barrel gets too hot, instead of having something with a pencil barrel that feels like a soldering iron after 5 rounds.
Point is, you want that rifle for lots of effective practice in the first few years, not just for hunting. A .22 is a good start, but won't get you to 400 yards.