I'm a big fan of Savage firearms. I have 6 actions, 12 barrels and 7 stocks. Plus the Savage 99 in 300 Savage. I would recommend the Stevens Model 200 as an entry level gun. It will give you more options in aftermarket parts than an Axis (the new name for the Edge) will. The Stevens is just a stripped down Savage, so it will take the same stocks and upgraded parts as a comparable Savage action. If you're a big guy you may need to get a stock with a longer than average length of pull, so aftermarket support could be an important consideration.
As far as "big" calibers go, they don't start getting big until you're shooting bullets over 225gr at more than 2800.

The 45-70 doesn't qualify as a big gun in any rifle. Put it in a handgun, and now we're talking!

Don't start out with the big stuff though. the 270 and 308 would be great choices. work your way up to the bigger stuff unless you've been shooting the 308/30-06 class rounds for a while. Big guns take a lot of practice to get good with, and they certainly aren't cheap to shoot.
Andrew