I have all three. Any of them will do the job of the others (7.62X54, 7.62x51, 7.62X63) I think a lot of the "Kick" people mention with the Nagant is due to the bad ergonomics of the stock. If you prefer a short action, I would give the nod to the 308, long action the 06. With 150 grain bullets, the higher pressures of the 308 will give slightly higher velocities using speer data, equal using lyman. With 180 grain bullets, the 06 and its larger case capacity will give higher velocities. At all practical ranges, all are equal though. The mosin has surplus ammunition very cheap, yes. You will have a hard time walking in to your local wal-mart and finding ammo for it though. I have no idea what other firearms you own or are planning to purchase, but the M1A, HKG3, CETME, etc, all shoot 308.
Scope mounting on a mosin is a bit of a chore also, and accuracy of some is questionable at best, while others are superb.
IMO, a 308 is a good platform. With reduced loads, the smaller capacity of the case leads to a bit more consistency, although with time, you can develop a good reduced 06 load.
Stay safe.