I know I'm a bit late responding to this, but I just wanted to chime in and say that in my experience, a Disney trip is the BEST vacation to take with a baby. We took DD to WDW when she was 15 months old. We weren't particularly interested in Disney before that, and we only did it because my in-laws offered us free use of their Orlando timeshare. Neither DH nor I had been to WDW since we were kids, and I really wasn't sure how it would go, but it was truly magical. There was so much for her to do and see, Disney is incredibly baby friendly, and she absolutely adored the characters.
The trip was wonderful for us because it was wonderful for her. DH and I traveled extensively in the 9 years before we had DD, and we have taken her on several trips to various locations since she was born (she's two now). That first Disney trip ranks among the top vacations that DH and I have EVER taken (and our previous trips include Europe, Japan, the Caribbean, and Hawaii). We loved it so much that we went back to WDW 7 months later (last May), and we are going on our first
Disneyland visit very soon! My goal is to make it back to either WDW or DL one more time before DD turns 3 next summer.
That being said, I don't have DL-specific tips, but I do have tips from our two trips to WDW. My number one tip is not to plan too much in a day, and to remember that you will probably not accomplish nearly as much as you think you will ahead of time (babies slow things down a lot!). I'd advise a relaxed, go with the flow type of attitude, and just allowing the magic to happen. Another tip is to plan a character breakfast on your first day. That way, you'll know immediately whether or not your little one likes the characters, and can plan the rest of the trip accordingly. And, like others have said, definitely use the baby care centers!
Have a wonderful trip!