My SAR team, which is youth-led, runs auto-aid with the local fire department for all structure fires and other major calls. We also share one of their stations, and they let us ride whenever we want. They do about 80% medical and 20% fire.
Like was said before, what you're allowed to do greatly depends on the rapport you build with the people you're riding with. If you can earn their trust, they'll usually be happy to let you play. Remember that your relationship with them starts at the station, not on the calls. Offer to help clean up around the station, pay for your dinner, bring some cookies over at the beginning of the shift, and just show them respect in general. Be enthusiastic too, and let them know what kind of skills you'd like to work on.
Advanced First Responders (90% equivalent to EMT-B ) from our team that build trust with the firefighters are usually allowed to do quite a lot. I have yet to ride, but friends of mine have run calls, done CPR, splinted, controlled bleeding... etc etc.
As far as fire scenes, we usually set up RIT, roll hose, change air tanks, keep bystanders out of the perimeter, manage evacuees, and do scene security (babysit it until the investigator comes out).
