Wow, a lot of people are spending a lot of your money.
My opinion on what a three gun set with the most value would be;
1. 10/22 - cause they're awesome and you mentioned you'd like one
2. used 870 wingmaster in 3" 12 gauge from the EE or Epps
3. Enfield sporter from the EE or Epps
You might have to get the 10/22 new because people who own them usually love them. But you could probably get all the above for under $700. If you want to go up a step look into a Stevens 200 in 30-06, 300 winny mag, 270 or 7mm Remmy mag. It's under $400 at Cabela's and has lots of upgrade potential, but don't forget mounts, rings and scope!
With the above, you shouldn't have a problem with ammunition, parts or expertise availability and they have loads of after market add ons.
That should leave you lots of money for gun club dues, travel, ammo, hunting clothes, reloading equipment, etc, etc, etc...