The airline and class does not matter.
So do people have multiple cc's, and use whichever airline has the best deal for the location they desire?
Yes.
In your case, I would assume that you are most likely to be flying AA more than anyone else. I'd suggest enrolling in their FF plan, obtaining an AA credit card, and registering the card in their dining for miles plan (no charge). I'm not sure what credit card offers are available right now, but about 18 months ago I received 30k miles, no fee for one year, $100 off purchase of travel on AA, and one free bag on AA flights. When the year expired, I called to cancel the card, but they offered to refund the annual fee for a second year and give some bonus miles, so I've kept it. You should be able to find something similar.