Does Islam allow us to have relationship with a girl? What is the appropriate age to do so.

According to Islamic Shariaya when a girl becomes 9 years old ( according to Lunar calendar) and a boys becomes 15 then they are BALIGH ( adult) not in this condition both should not have any type of relation with any na mahram which could lead their relation to a Haram act. But they can talk each other do work each other as long as they follow the rules.
Now what you are asking has no place in Islam. The only thing I can say is if any boy likes a girl and a girl likes a boy and they have decided to become life partner,s they can do Nikah or if possible do Mu’ta with the permission of girl’s father and then they can talk each other and understand each other then after that can do Permanent Marriage.