Of the two cartridges, I'd vote for the 30/06 as a deer-specific cartridge. The .300 will, of course, do the job, but the added power over an '06 is not needed for deer. For someone new to the sport, the added range should not really be a consideration either. Being familiar with your '06 and it's trajectory will allow a hunter to cleanly harvest deer to 400+ yards and it takes a practiced shot to be confident and capable of that task.
As far as the platform you've mentioned, I am not a fan of that rifle. I think that for slightly more money, there are better platforms out there. The Rem 700s and Model Sevens may be excellent rifles, but the 770 has exactly zero in common with either of those fine actions/rifles.
Marlin produces a rifle in that same price range, the XS7 that has garnered lots of good reviews. The Howa 1500 can be found for slightly more if you search around and I feel it a much
better platform. The Mossberg ATR 100 would also be a better choice. Not everyone likes the same rifles, but buying simply for "maker" and price might not be the best idea in this case, IMHO.
There are also a whole lot of quality used rifles in that price range that I would also recommend above the 770. The Rem M700, older post '64 M70s and any number of Ruger M77s & Savage 110s in their various renditions would all make a very good choice for someone. The '06 is not hard to find in any of those models.
Best of luck with whatever you choose and do not forget to get a quality scope for on top as well!!