There are many ways to get competitive with robotics. The best way, is to start small and local. Get others who are interested in robotics and set a common goal. For example: Following a simple line, moving an object from one place to another, sumo robotics, etc. As the challenges get more complex, you should think more about national competitions. FIRST is a national competition targeted toward many age groups.
Here is more info about FIRST:
Here are a few videos that can give you perspective of the challenges and robotics.