I'd probably build off a 10/22 since it is probably the most popular .22 rifle out there.
Assuming money isn't a factor and building off a complete, factory out of box rifle:
1. Volquartsen barrel. I have the Stainless Steel Match 20" match barrel w/ compensator. I'd probably get the same again, but with flutes.
2. Kidd Trigger. (
http://www.coolguyguns.com/match_trigger.htm). If you buy this, there is no need to buy Volquartsen internals to lighten the trigger pull, have the auto bolt release, and extended mag release features.
3. McMillan Sportsman Team Challenge (STC) stock.
4. Assuming going with Volquartsen barrel, get a Volquartsen Barrel Mounted Scope base.
5. A decent piece of glass. Find something along the lines of Burris, Weaver, or Leupold.
#1 will probably cost $300, #2 $300, #3 $600, #4 + #5 anywhere from $250-$400 depending on the scope. So including the $175 for the initial rifle, this would be a pretty expensive project at around $1700.
If you want reasonable parts recommendations for the 10/22?
1. Any Green Mountain or Volquartsen barrel should be ok. I haven't purchased GM before but everyone says they're the best bang for the buck. I've owned two VQ barrels (THM and SS Match w/ comp) and I loved them both. You will spend anywhere from $100-$300 depending on what you go with.
2. Get the Volquartsen Auto Bolt Release, Extended Mag Release, and Target Hammer. This should only be about $50-75 or so.
3. You can get the Hogue Overmold Stock, which is a bargain stock for $75 or so. I'd look into the Bell and Carlson Anschütz stocks. I'm thinking about getting one. It runs for about $200.
4. As far as scope bases, I like the barrel mounted design. It's just a personal pref. Some people argue both ways for receiver or barrel mounted scope bases.
5. Don't skip on glass. Your scope should be a significant purchase. No point on spending so much money on a rifle and then put a POS scope on it. I went with a Burris 4-12x32 Compact Scope with Side Parallax adj for my 10/22 and Leupold QR rings on a VQ barrel mounted base.