While it is possible to hire a professional genealogist, first do as much as you can on your own.
First of all, get a large piece of paper (or cut open the sides of a paper bag). Then write down everything you know about your father’s side on one piece of paper and then your mother’s side on the back – or on another piece of paper.
Now try to interview anybody in the family older than you – especially previous generations. As you talk to them, you are going to write on the piece of paper. So why do you need a piece of paper instead of the internet or a computer? Very simple. As you are talking to them, they will tell you interesting stories, or maybe the name of the city where your family came from or the phone number of a relative you might want to talk to or something else unexpected and you can write that in the margins. Computers don’t let you write in the margins. Ask them for addresses and phone numbers of other relatives.
The next thing to do is a DNA test. Spitting in that test tube might help you find some more relatives.
It is entirely possible that somebody else in the extended family may have made a family tree that includes you
Once you have done as much as you can on your own, then you can try the internet or hire someone to do that for you. And while the pro’s can go pretty far back, the question is what are you trying to accomplish by knowing your ancestors names? You will learn far more about things that matter by interviewing relatives.