I don't know too much about the 783. Remington's previous economy gun, the 770, was almost universally condemned but they may have fixed the problems in the new gun.
I'm a big fan of Savage rifles, on the other hand, and their Axis line always gets lots of recommendations whenever someone is looking for an inexpensive, entry level rifle.
As far as caliber choice, I would not recommend the 7mm mag for a new shooter, or anyone else for that matter unless they needed a long range rifle for elk or larger game. The .30-06 and .270 are quite close in capabilities. The .270 is a bit flatter shooting, while the .30-06 can handle somewhat heavier bullets. For deer size game either one will do.