You can. Genealogical research isn't rocket science--it takes order, never assuming and verification. There are plenty of web sites that have instructions: just google something like how to do genealogy. My 2-cents is to stay away from the Trees.
It is often just as easy for someone to do their own family tree than to have someone else do it. As @Fairfield says, there are many websites that are eager to help you. It's really just a question of sitting down and taking the time to put in what you know. The basic rule is to start with yourself. Open a website - FamilySearch is one that many people use and it is free - and put in your own name and that of your parents. Add dates for birth and, if applicable, death. Put in where your people have lived. And then let the search engine suggest records that match. Often as you see Census and other records they will fit what you already know about your family. Then you build on what you have learned to go back a bit farther in the past.
You may hit a brick wall at some point. Many of us do. That's when you want to reach out and ask for help.