30-06 has an average of 170 FPS more velocity in same weight bullet as a 308. If you load a 150 grain bullet in a 7 Mag and a 30-06 using IMR 4350 powder, the 7 Mag uses about 2.5 grains more powder and is actually 11 FPS SLOWER than the 30-06. (Hodgon's Reloading Manual). The 30-06 and 7 Mag actually swap back and forth for velocities with same weight bullets depending on powder,but the difference is amazingly small. An advantage the 7mm has at 150 and 160 grain weight bullets is a better ballistic coefficient and sectional density than the same bullet weight at .308 diameter. That might get you a handful of extra yards of usable ballistic energy.The 30-06 starts to pull away at 160 grains and heavier. Thing is though, about heaviest in 7 Mag is 175 grain while 30-06 can go to 220 grain. Now, that doesn't help with normal deer hunting,but it does show that the deer wouldn't know the difference once hit. Each gun is as accurate as you choose to make it. My 30-06 can group 5 shots a quarter can cover because I hand load for my rifles.
7 Mag isn't as common a cartridge as 30-06, but is common enough;just not as many loading options. Belted mags like the 7mm headspace off the belt,so there can be more throat erosion and less accuracy--but is a minor thing in a deer hunting gun.
I grew up with 30-06,bought my first one when I was 14 and have had one since. If I had a 7 Mag or 308 first I would have probably stayed with that caliber.
I would concentrate on getting a good rifle in any of the three,use a 150 grain bonded core bullet like a Nosler Accu-Bond or Partition,Sierra Game King, Barnes Triple Shock, Hornady InterBond or SST, even a Remington Core-Loct.
308 has mechanical and logistical advantages over the 30-06 because of size/weight of the cartridge (works better in semi-auto and machine guns) and statistically 'might" have a slight advantage in accuracy if you compare thousands of rifles. It is the cartridge for long range in the military because it is what the Army buys (338 or 300 Win Mag easily outdistance and outperform it).
I reload for 223,243,30-06 and 300 WSM.