For simplicity, excellent accuracy, dirt cheap to shoot, good factory ammo if you just want to plink, easy on the shoulder and barrel - 223 Rem. Get a 9 or 8 twist so that you can shoot up to 75gr Amax. Not the best wind bucker (same like the 308) but certainly lots of fun out to 1000yds.
The 6BR requires loading, is wonderfully accurate, more expensive for components, can be set up for all ranges. Not the best wind bucker for LR but better then the 223.
A 22/250 with a fast twist barrel will be 200 to 300fps faster then the 223. Will reduce wind drift a bit but much harder on barrels.
The 6ppc is also an amazingly accurate cartridge but like the BR, must be reloaded. Not ideal for LR but superb in the shorter distances.
My guess is that your range doesn't go beyond 300yds. If this is correct, the 223 will do everything you want. In fact, I would recommend the Savage 12FV as a great starter rifle. Bed it, reload and you have a 1/2 MOA or better rifle for under $650. Lots of extra money for optics and ammo.
In time, you can put on a match barrel/restock once you have an idea of what you want to get next.
Jerry