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:

Link to their Page for registering a team

FIRST Tech Challenge (FTC) is a robotics program for middle and high school students. Student teams design, engineer and program a moderate sized Tetrix robot to execute skills and challenges adapted from real world robots of similar size. click here to enter site

Here are a few videos that can give you perspective of the challenges and robotics.