The term dating is not used in Islam.
As for how to go about finding more about the person you are interested in:
Girl, Tell your parents that you are interested in getting to know and potentially marry a certain Muslim boy, your dad (or both parents) will then go to the person house and talk to the person as well as their parents and say that their daughter is interested in getting to know their son better.
From there the boy can visit the girl's household with permission of her parents and be supervised by her Mahrams (Father or Brothers) and they can get to know each other like that, keeping it Halaal with no physical contact between the two.
Boy, Basically the same way, only he goes to her house. With Regards to going out together:
The boy can go out with the girl's family when they are doing other activities and he can speak with her with supervision from her Mahrams, keeping it casual and Halaal.
The Girl can go out with the boy's family and get to know the boy better but should be accompanied and supervised by her Mahrams to ensure her safety as well as to ensure everything stays Halaal.
For the purpose of the post please do not feel offended by me saying Girl and Boy, it can be a Female and Male of any appropriate age.
I am not married yet as I am focusing on my studies and would like to be financially stable to some degree before getting married, so please keep me in your Dua’s that Allah will grant me a Good compatible Wife, Insha-Allah.
Have a blessed life and remember that Allah is always there for us, it only takes a moment to appreciate what he has given us and to ask him for anything.