I think it comes down to your training. I know we just spent a half a class on ICS and what to do if you're the first one on the scene. Just follow your NREMT patient assessment grid with a few additional steps thrown in. The only steps you might want to add are radioing back to the rig before it gets there to give them a heads up of what you find.
BSI, scene safety, NOI/MOI, # of patients, etc...
If you show up on a 2 car MVC with 3 patients, get a call back to dispatch/the rig and let them know. "This is unit 5 to xxx county, I'm on scene and we have a 2 car 10-50 with at least 3 injured, roll 2 additional units and the FD for extraction/traffic. " then maybe relay that to the rig so they know what's going on and start triage.
Same with a Medical, just follow your grids. BSI, safety, NOI, # of patients, c-spine, additional resources, then get into your general impression and make patient contact. "Hey, my name is XYZ, I'm with the Rescue squad. Dispatch said you are having troubles breathing, is that right?.....etc. Then just take it from there. Or you got called for SOB and it turns out to be asthma, give the rig a call and ask when they get there to bring the portable neb off the rig when they get there.
I guess I can't give too many tips or tricks since I have very little experience myself, but I'd think first on scene is the same as most senior with the rig right?