Scamper,
If this is your first gun I'm gonna have to recommend a good single shot such as the savage Mark I. There is a lot that can be learned for shooting a single shot rifle for a couple seasons. Now, that being said, I personally own a marlin 60 and ruger 10/22 and love them. I know that they are two of the most repeated recommendations when looking for a good 22 but there is a reason for that. They've probably killed more squirrels and rabbits than any other two models. For hunting situations with a 22 at shorter distances a semiauto will offer that quick second shot. So I honestly recommend them as well. Just a lot of gun for not a lot of money.
If you go for a bolt action, I urge you to give the 22 mag or 17 hmr a look. They're slightly more expensive but are far superior at longer ranges that often faced at the end of the year when cover is sparse and squirrels are running on the ground.
Hope it helps bud
Indy