Having flown SW with lap babies probably 40+ times over past 10 years, I agree with almost all the advice already offered. Make sure to bring the official birth certificate (copy is fine)--not the hospital one. Board in A (if you get A passes) or at family boarding and head to the back and grab a row of 3 behind the wing. Odds are you will get to keep all 3 seats. As for routing, nonstop is preferable. If you have to stop though, I would recommend booking something with a 1-1.5 hour connection time rather than a direct flight where you stop and don't get off the plane. In my experience, it is better to get off the plane, have access to a full size restroom, opportunity to get real food, walk around, etc., rather than sitting on a plane on the ground for 45 minutes while they turn it.

Couple more tips: we usually check our carseat and take a small stroller through security and use it to get around the airport. Then we gate check the stroller -- you leave it on the jetway and pick it up on the jetway. If you book a connection, you can specify whether you want it at the connection point or at the final destination. SW is great about this. Some people like to strap a car seat in on the plane, but I don't like hauling the heavy/awkward car seat through the airport so I just check it at baggage check.