We took our DD for her 1st time when she was 9 months old (Dec 2001) and stayed at the poly. It was a great trip. Poly is a great place to stay with a baby. At the time we had this really cool Instep backpack stroller that we used and it was great for us. When the parks got too busy we just changed it to a back pack and walked out with out worrying about crowds (even during a pararde).
We brought baby food and I was still nursing. (that made life easier, you have everything you need with you all the time.) We also brought cherios and other snack food for her. We were able to do everthing you can do with a todler and we all had a great time (and got some great pictures.) Her 1st ride was Dumbo and now (she is 3 1/2 and has been 4 times total so far) she has to ride Dumbo 1st every trip.
Our next trip will be March 2005 and we will have a almost 4 year old and a 7 month old (4 weeks old today, that means we have only 6 month until our next trip!) So back to the baby stuff.
Town car services do provide car seats free of charge. We had a problem once with one compmany not having the seat (both ways to and from the airport.) They did find one, but it took some time.
Our infant car seat can be secured with the base, or just the seat belt if needed and we plan to bring it and do that. I havent decided if we are going to buy the baby a seat, we flew 2 times before with out one for her, it worked out fine. Air fare is starting to add up. We brought the infact car seat and checked it. I wouldn't depend on the town car having a rear facing car seat for infants, they mostly have them for todlers.
What it is like traveling with a baby that age really depends on your baby. I will tell you this, relaxed parents tend to make for more relaxed babies!
Good luck, I am so happy to not be pregnant anymore! I was really ill for the 1st half of both pregnancies. I hope all goes well for you. BTW I love your count down clock. I clicked on the link and I was just tickled pink. My baby's name is Lily. Lilypie. How cute is that? It is great for keeping track of the time. I never could remember how pregnant I was until I got the email from babycenter telling me how many weeks pregnant I was. Then I turned my trip countdown clock to a due date one here.