22LR is probably your most reasonable choice.
22LR can be had in various quality from match grade to bottom of the bucket plinker rounds. The problems with most other rimfire rounds is availability and lack of high quality consistent muzzle velocity. 22LR is bad enough in this regard, but there is no match grade 22WMR and the .17 rimfires are all over the place. Sometimes you can find ammo, sometimes you cannot. The general rule of the more $$ you spend the better the ammo really applies here.
Also, if you think that you might ever want to go to a rimfire match, they are almost always limited to 22lr as 22WMR and .17 will chew the steel targets up and are not allowed. Also, since 22lr is so popular there is a lot of data on it so dope can be figured pretty quickly.
Since you have stated you are looking for a bolt action, (good choice, the 10/22 platform is a never ending $$$$$$ rabbit hole) I'd recommend a Tikka Tx1 or CZ457 (specifically the 457 as it corrects a few issues of previous models) if you can extend your budget to $500.00. If you cant extend your budget then a savage will work with caveats, I've seen too many extractor and magazine problems with savage 22lr to recommend them. On the plus side, every savage I've shot with has been accurate, just not reliable. Ruger rprr is a crapshoot for both reliability and accuracy, but every body I've seen start out with one has gone to a CZ, tikka or vudoo eventually. Ruger american I'd skip as it has extractor problems--I've personally know two guys that carry a small slot screwdriver to extract stuck cases in the barrel, even after one of them had sent it back to ruger 3x for this problem.
