I am against paying a lot of money to get my hair done and actually usually cut it myself, but I recently went to an Aveda salon (they have different names per different city but they're owned by the same people and have the same methods). It cost me $40 (the price goes up from $40 as the stylist has most experience with the company). The great thing about the salon, though, was included in the price, you get a neck/shoulders/scalp massage, shampoo/condition, cut, and style. A lot of places charge extra from shamppo/condition and style, plus you don't get the massage elsewhere.
The price is fairly reasonable if you have the money. It's definitely an experience even if you're only 13. Regardless, I would suggest that you NOT go anywhere that looks really cheap, because they may not do a good job. If there are any beauty schools around, you may want to look into going there. They are cheap (usually less than $10) and the people are learning, but they do have experienced teachers working directly with them when you go to have your hair cut.