Hi Andy,
I would begin with SolidWorks Tutorials available from the Help file. Lesson 1, Lesson 2 and Lesson 3 are a good place to start. Then I would do lessons that has to do with machined parts or assembies. If you are working in sheetmetal and need to create a flat pattern, then step through the Sheet Metal tutorial. In the summer, work through the tutorials in plastics, mold design, animation and photorealistic rendoring - right now you dont have time with the competition deadline fast approaching.
SolidWorks made a series of robot tutorials with movie files at
www.solidworks.com/vex to help you. There is a .pdf file to download to introduce simple parts and assemblies.At the bottom of the page you can download all 14 tips to design better robots.
The biggest tip I can give you in robot design is don't create any component that you can obtain the model file from
www.3dcontentcentral.com or another source.