I have flown Spirit a few times to Disney from Atlantic City, NJ. Basically it is a no frills airline. Be sure to add in all anticipated fees before you book, as sometimes Spirit is cheaper and sometimes not. They have a $9 club which saves you on flights and baggage ($50ish per year).
If you want to sit with your kids or travel companions, it is a minimum of $10 each way per seat. They charge for carry ons. Seats are very small. I have had no specific problems with any of my flights. I did have to call customer service once as they changed my flight, which then resulted in time change, plane model change and the computer reassigned my seats. I was notified of nthe changes, so it was no big deal, except that the computer reassigned our seats to the exit row, which is a no no with small children. Called customer service (which I am pretty sure was not someone from the USA). Spent a lot of time on the phone as the customer service agent assured me this was fine. I asked to have seats reassigned and he could not figure out how to do this in the computer without losing the money I already spent. Called back later and got a different agent and she took care of it.
Checked bags have a 40 pound limit (industry standard is 50#). Paying $36 for an additional bag is not a big deal, but if you are renting a car be sure to figure in the possibility you may need a larger vehicle to fit additional luggage, which could really add up depending on when you are traveling.
I would fly Spirit, but only if I was saving a significant amount of money over a competitor airline.