Somewhere around thirteen is a good age to start liking someone and consider them your boyfriend/girlfriend, and maybe hang out after school. However, I believe that it's too young to be going to dinner and the movies at night. Fifteen or sixteen is a good age to start doing that, but still to young to get too serious. :)
13-16 depending on maturity, because dating involves sex and sexual situations. If you are too young it is easy to get manipulated into doing something you may regret in the future. You need to be old enough to understand the situations and know what you are and aren't willing to do based on your moral code.