We had no trouble at all with ticketless travel. I had my print out from the reservation that I made online but she did not even look at it.
All she needed was my name and then my passport I gave her for ID.
Since you have never flown before I should also warn you that SWA does not assign seats. When you check in they give you an A, B, or C boarding pass and then you line up in those lines and when they let you board the plane you just pick an open seat.
I was very concerned about this last year when we flew because there is 6 of us but it worked out just fine. We were able to get A boarding passes when we checked in (and we did not arrive very eary - maybe 1.5 hours before take off) and we sat 3 & 3 there and back.
It is a very efficient airline and they are usually right on schedule. There are TV's on board but it is only a 2 hour flight from Buffalo to Orlando. They do give you a snack and drink mid flight.
As I said before, I was a little worried about this no frills airline but I was pleasantly surprised and will now fly nothing else if I am flying within the US.
Too bad we did not have such an airline within Canada! I am still a little sore that 6 of us can fly to Florida for $451 US and it is going to cost us almost $4000 for fly from Toronto to Vancouver next summer. There is just something wrong with that math.