Of all the budget rifles I've come across, the Ruger American is a very sound shooting rig. I have experience with the 22-250, .243, .and the 270 win. all are good performing cartridges, each with their own characteristics. Ruger also makes the American in .223, .308 , 30-06 and 7mm-08.
The 22-250 is a very high velocity low recoiling round, perfect for shooting varmints and target shooting. The 22-250 will do anything the .223 will do, but with higher velocity and more bullet options to choose from. 22-250 isn't an expensive cartridge to shoot, and factory offerings can be had for under $20 per box.
The .243 is good for both varmints and big game hunting applications. Great for deer and antelope, and with the right bullet it can be used for elk sized game, but that would be pushing it for a new guy as shot placement would be critical. I have personally known people that have taken both elk and bear with the .243 win, but it's not the optimal choice for such. Also not an expensive round to shoot, I see boxes at Walmart for slightly under $20 per box.
The .270 will do it all, it's a nice flat shooting cartridge that can be used to hunt just about anything on the North American continent. .270 ammo is usually in the $20 range per box, but a little more expensive than .243 win..
With half decent optics, all three of the above cartridges / rifles will easily produce great accuracy at 200 yards and out.
A Ruger American is under $400 NIB. A Vortex scope of decent quality for shooting 200 yd. distances can be had for under $200. So for a total of about $500, you can be shooting a nice high powered rifle. Ruger will warantee their rifles for life and they have great CS.
GS