The Ruger 10/22 is a great and immensely fun gun. I have had several over the years, most recently a Talo edition I picked up for under $290 (OK price.. It's been higher and lower since I bought it).
The first rifle I bought for my daughter was a Marlin 60 which can be acquired for around $200.
This said, I have a fondness for bolt guns. I enjoy all types of actions, but something about cycling the action is just immensely fun for me. The only thing I can liken it to is a manual shift versus an automatic
. I don't know of a really excellent 22LR bolt action gun though, and for a first gun, I think it's a good idea to go with 22LR. You could nudge up to .223, but many indoor ranges don't allow any rifle rounds except 22LR. And unfortunately in this day we can't walk over to the lake and practice anymore (at least not where I live now).
BTW, I've also had a few Remington 597s, but there appears to be some grave issues with their safety.