Also checkout First Lego League.
It's a great program. This is my Son's third year and our second year coaching his team.
The season is in full swing right now so you could probably go observe (possibly join) a local team. And find out when their competition is and go watch.
Here's a video of what they could create.
The FLL program consists of core values, a project and the robot game. Robot game is the most fun but they have a good time coming up with a project idea and learning about core values (work as a team, do the work, etc).
If you want to start your own team you can get donations/sponsorship from companies to pay for your startup costs. 2 or 3 robot kits, some extra parts, T-shirts, the FLL kit (mat and mission pieces), supplies to build the board, FLL fees, etc.