Good money? Depends on your definition, but hanging onto a hammer isn't going to set any income records. Plus you are beating the snot out of your body constantly. If I could transport my 34 year old brain into my 17 year old body when I got into the trades, I would do something else with my life than be an idiot construction worker.
If you want to get into it, I wouldn't bother with school. Other than classes unrelated to the actual trade itself, you won't learn anything that can't be learned while getting paid on the job.
Bigger companies will have dedicated trimmers, roofers, framers, tile guys, guys who do wood floors, etcetera. BUT, there's plenty of small contractors who do a little of everything. My company does nothing but cabinets, but I do find myself helping out contractor buddies once in a while framing when they're short on help, (and I don't have much going on). Not my favorite job, too much like real work.