No. By that I mean you should not be going out somewhere on your own. I think it is ok to have a boyfriend that you hold hands with and give a quick kiss goodbye to while at school or at group activities where a group of you may go to a movie or skating.
It's not healthy to date at 13 because then you are focused in dating too much and at that age it's not technically "dating" it more like getting to know each other everyday. Plus, "dating" can make you not pay attention or put you away from more important and long terms things that can never end in life such as studying for your education, spending time with family and friends. I recommend you date when you are mature enough to handle it and know what dating is, cause and effects and that you should focus in permanent things such as studying. :)
Group dates could certainly be fun but one on one dating should not happen till one of the two of you can drive where you want to go. Otherwise you will have to ask you girl/boy friend " Wanna go to the movies with my mom and I?" Also at 13 a lot of girl's dads still want their babies not grown up and not hanging out with guys.