Then a 10/22, especially a takedown model, should fit the bill. Note that a takedown model will be slightly less accurate than a non-takedown, so if you're going for super precision shooting, then don't go the takedown route.

A few other things to consider - if you're planning on only buying ONE gun, you might want a slightly higher caliber, such as at least 9mm, in case you need to use it to defend yourself - sticking with rifles, the Ruger PCC is a good one (although heavyish), and is also a takedown. Obviously if you're comfortable in getting a first rifle for learning the basics, then adding another rifle to your lineup later that's got more bells and whistles, this won't apply to you.

Another two considerations, especially if you want to be more on the DL, is storage and transport. Your state/county/city might have storage laws that you may have to comply with, making it slightly harder to store rifles than pistols. And clearly if you have to put a big rifle case in your closet it'll be more obvious than a tiny pistol safe. Regarding transport, it's pretty hard to covertly carry a full sized rifle around, since the cases are so much longer, whereas takedown models such as the 10/22 or PCC are much easier to fit in a (tall) backpack or tennis racket bag. Pistols are obviously much easier as well.