Do I have to use the track mat that came with my train set?

You do not have to use the track mat supplied with a starter set, ultimately, it's your layout and how you play with it is up to you.

What a track mat does do is protect your models from accumulating dust, fluff and debris from the carpet which over time can affect the running of the motor. A track mat will also show you what it is possible to achieve in a relatively small space. The track sections and buildings printed on a track mat are the same sizes as the full-size models that you can place on them, so they provide an excellent track plan for anyone without their own ideas.

There is no written rule to say you should or shouldn't use a track mat either on or off a baseboard. But how about using one to start with when you haven't a lot of track or buildings to provide scenery and replacing it as your collection grows with a more permanent structure?