I agree with the other posts here regarding the bolt actions vs the Ruger 10/22. I built a few Ruger's for fun, for an education, and then used them for benchrest shooting. Best I ever did was a 20x-250 with Wolf ammo. A buddy of mine had a Ruger #3 in 22LR and he hit a few 25X-250's with that. I've read of the 10/22 with Valquartzen stuff that will do better but the price goes up too!
Other option: A T/C Encore with a match/accuracy barrel (eabco.com) and target stock will allow you to get to your next cartridge by just purchasing another barrel (for around $400), and another for varmints, and another for Elk, and... but I degress
If your going to migrate into larger cartridges you may want to consider the action your moving to. If your going to go to a bolt gun then by all means train on a bolt 22LR. If your going to use an AR style for target shooting/competing then maybe you would want a semi-auto 22.
If you can find a used 10/22 ($200?) to set up for target then it will need the following to be competitive:
1. Trigger (Timney or other, add $100) or polish the sear on stock trigger.
2. Stock (Heavy barrel requires a stock with a larger inlet for bull bbl, I like Fajen's, adjustable cheek and length of pull - $129.00)
3. Heavy Barrel (many good barrels for around $100-150+)
3a. Optics, make sure you get something that will have parellax down to 15 yards if your going to shoot benchrest indoors (15 yards).
For optics you can go inexpensive for a good one ($300?) or buy a really nice scope (for that 260 Remmy 1000 yd rifle down the road?) and be able to move it to your higher caliber when ready to upgrade.
So yea, the Ruger will run around $800-1000 (depending on optics) to shoot sub-moa. You can do it in stages and still practice if upgraded in the above order. The Fajen stock will work with the original barrel but will be required for a new bull barrel. The optics will have to be added when you add the barrel because there are no sights on the bull barrels.
One thing this rig will do that the bolt or single shot gun can't; pop off 5-penny sized targets at 50 yards in about 5 seconds or less!! The 10/22 has a bit more "fun factor" then the others.