OK, is this a good summation of your situation:
Origination airport:
1. LGA
2. JFK
3. EWR (Distant third)

Goal:
1. Free domestic flights
2. No other goal identified. (i.e. various elite status benefits like upgrades, waived fees, priority access, mileage earning bonus etc)

Approximate annual flying:
1. 2-3 trips to FLL (8550 to 12775 miles/yr)
Will you consider MIA?
2. 1 trip to europe (lets call it a conservative 15000 mi/yr)
3. Other sporadic travel that might net you a coupe thousand miles.)

A reasonable assumption here is 25K miles per year, which will, by itself, get you close to a free domestic trip on many programs, and will qualify you in the first elite teir on those programs. Clearly, focusing your travel with one airline or alliance would be worth it with this pattern.

I will let other with more experience flying out of LGA and JFK address the specific airline recommendation.

But with yoru flight pattern, you should easily be able to meet your primary goal, and if leverage a few other things, like credit card promotions and miles for spending points, you can do better than you think.