Premiere is great and if you're used to Final Cut the jump over isn't too jaring. I would also suggest getting a Dropbox account mostly because the
file request feature makes it super easy for students and parents to submit footage and pictures from personal cameras or phones. It'll be a chore to keep it organized and maintained, but you'll be glad you put in the effort when you sit down to start editing at the end of the season.
As for cameras, having a GoPro is good for some footage, but I wouldn't rely too heavily on that. I would try to convince the team to invest in a decent camcorder, (I have been using
this camera for the past few years and it's been great, but that guide* has a few other options depending on budget.) In my experience a small camcorder like that has a good ratio of decent quality footage/ease of use. I'd also suggest a simple external microphone (something
like this would work fine) if you want to hear anyone talking over general shop or competition noise.
Most importantly, like Francis said, the more footage you can get the better off you'll be. Encourage your team members to capture footage throughout the season on whatever device they have ready access too, and if you keep it organized it'll be much more manageable to put something together at the
end of the year.
*Disclosure: I work for TheWirecutter, but have been using that camera since before I started working there. I linked to that guide because I think they do a better job explaining the camera than I could here.