Elliot,
Your head will hurt when you start looking into scopes. Lol.
I have plenty of Rem's and Winchesters. Both are good. Winchester is just better.
Handle them both first. Put them up to your shoulder, work the bolt, pull the trigger, work the safety. Look at the actual quality between the two. One easily wins that department, but you'll have to decide for yourself.
A .243 will handle plenty of big game, including bear if you use the right bullet. Don't let that bother you. You'll also be able to shoot it a distances farther than you can probably see, so don't worry about the range either.
If I was spending the money, I'd go to the gun shop, spend $850 or a Win M70 Featherweight in .243. Spend $400 on a Leupold VX-3 2.5-8 scope and rings, and shoot the hell out of it. 
