If you could stretch your budget just a little farther to $400, the Savage Mk II FVT will keep you challenged for a lifetime. It has a bull barrel, Accu Trigger, and Williams peep sights which are head and shoulders above the Ruger 10/22 sights. The Savage is for my 6 year old, I bought a CZ 452 for my step-son when he was 13, solid rifle purchased new for $330. The stock even has a decent figure for a $300 .22, shoots better than he can. The CZ is full size rifle designed for cheap training, to mimic CZ's large caliber rifles. The Ruger and Savage comparatively are smaller.
I lucked out and my first rifle was an Anschutz 1416, my brother and sister received Ruger 10/22's with Williams sights. The 10/22's are good rifles but the bluing and materials used are pretty cheap, both my sister's and brother's rifle needed constant attention to prevent rust. On the plus side the aftermarket for 10/22 upgrades and parts is outstanding.
Good luck on your search, Savage, Ruger, and CZ are all good choices for beginners.