Now, you will find a lot of resistance to the 770. They really are designed to be the cheapest possible rifle to manufacture. They tend to have poorer trigger pulls and with the barrel press fit into the action will likely have worse overall accuracy when compared to a gun like the Edge/Axis models using the barrel nut.
I would prefer the Savage myself. I have never hunted with the Edge/Axis models but they seem like decent guns, particularly for the price.
Now, while this does move out of your price range just a bit you may look into a Stevens model 200 like this one http://www.budsgunshop.com/catalog/produ… The Stevens 200 is a Savage action placed in an older mold for the synthetic stock. If you want a gun that you can potentially upgrade in the years to come this is it. The barrel nut design means that you can easily have the barrel swapped out for another caliber if you want to. You can also upgrade to a nicer wood stock if you get tired of plastic (you cannot upgrade the Axis stock because the magazine clips into the stock rather than the action).
On a separate note, if it is only a deer hunting gun, do you really need a 30-06? Now a 30-06 is a great round and is probably one of the most versatile out there, but it is a bit more than you really need if you are only hunting deer. If you really feel the urge for a .30 caliber a .308 may be a better option. They are slightly shorter (resulting in a shorter rifle) as well as using less powder to get pretty similar ballistics particularly in deer loadings say 150 grain bullet weights. You may also look to a .270 or even something like a .257 Roberts or .25-06. Any of these is plenty of cartridge for deer with less recoil and would do a better job transitioning into a flat shooting varmint rifle should you get the opportunity to hunt small critters from a distance.
Avoid the Remington 770. Personally I would rather have a quality used rifle than a new el cheapo.