You can use any CAD program you'd like. Autodesk Inventor is provided to all FIRST teams as is the only program eligible for the the Excellence in Design Award (sponsored by Autodesk).
Many other teams use SolidWorks, which FIRST teams can get for free by requesting
sponsorship.
Other teams use Pro/E, or any number of other 2D or 3D programs. I'd recommend 3D for FIRST, but your best bet at this point is likely anything that you can access and that your students and/or mentors have the best familiarity with.
For at least Inventor and SolidWorks, the included tutorials really are quite a good starting point. Also check in your community and at local universities for seminars or workshops, and try to get professional mentors who use this software. After you know the basics, the best way to learn is basically by doing, hopefully with an educated guide.