I've used Jet Blue before with my Yorkie. I flew from Texas to IL last fall. Remember to get your health cerfiticate too! Pup cannot board without it. You will have to prove you have water for the pup.
When I boarded I asked the flight attendent if there were any empty seats, she told me to take my seat and once everyone was on she would come and get me if there were a place I could have a seat for my dog. Sure enough she asked another passenger if they would use my seat - thankfully they were a dog lover and gladly switched. Griffin got to sit on the seat during a portion of the flight. He was such a hit - several ppl felt more animals should fly and reduce stress for those that don't like flying. I think Griffin got more service than the humans on the plane did.
Best advice: Call ahead, make sure they know you are boarding with a puppy. Show up early, have all necessary paperwork ready so pup can board. Talk to your flight crew - make friends with other passengers while waiting to board (you never know who might be willing to move for you and pup to be more comfortable) and last - just enjoy your new pup!!