I recommend the 10/22 as well. It is accurate, moderately priced, and modifiable. The biggest advantage is being able to shoot 500 rounds for $20. I can only shoot 50 out of a .223 rifle for that much!
Personally, I am not a fan of the factory sights on the 10/22. Fortunately, a company called Tech Sights makes better iron sights that can be installed relatively easily (you would need a punch set, a screwdriver, and some loctite.) That is one of the few upgrades that I would seriously consider for a new shooter.
I would recommend taking a class like the Appleseed events. They can teach you to be very accurate with iron sights. In shooting, the fundamentals are what really separate accurate shooters from inaccurate ones. Once you learn and practice the fundamentals, you can expand in a variety of ways fairly easily.