DD has been twice, at 5 months and to celebrate her 1st birthday. We're going back in April (she'll be almost 19 months).
We've had
great trips. I'll add the disclaimer that my family has been going to WDW for years, so there isn't any pressure about not getting to ride/see anything in particular. We're able to go with the flow for the most part. We do have a plan laid out well in advance about which park to visit each day, and an idea of the top 3-5 attractions/shows we really want to shoot for in each park, but beyond that we just go with DD's schedule & mood.
We always try to stick to her usual schedule as much as possible. Things don't always work out perfectly on time (that's traveling) but we do our best not to push her too far. That does mean taking time out of the middle of the day to head back to the resort for a nap & pool time. As a PP mentioned, you might find you need that recharge time just as much as your little one!
DD is extremely curious and loves to people watch, so entertaining her in restaurants has never been too tough. For squirmy ones, character meals can be great. Or try places like Coral Reef in Epcot or Sci-Fi Dine-In in DHS. They have theming that will likely keep little ones occupied. Eating at outside tables can also be great if the weather's not too hot. For the most part she's eaten off our plates, although we have ordered her some kid's meals when we've suspected she's particularly hungry. I do always bring snacks and things like cups of applesauce and cereal bars.
I've never utilized the Baby Care centers, but I've heard nothing but great things about them. When it comes to diaper changes, I use a tip I picked up from another DISer: Every morning I make up several changing kits. I take a gallon size ziplock bag, put in a diaper, a few wipes (I stick these in a sandwich bag), and disposable changing pad. Then when DD needs I change I can just grab a kit and head for the nearest restroom; no need to drag in the stroller or the big diaper bag.
I definitely recommend bringing your stroller from home, particularly if it reclines. You never know when your little one might need a power nap after just an hour or two in the park. Be sure to bring a rain cover if you have one; there's nothing worse than coming out of an attraction to find a little FL cloudburst has soaked your stroller!
The past two trips with DD have been some of the best trips we've taken. Having a little one does force you to slow down and really take a look around at all the little things people miss while commando-touring. (And hey, we used to be that way too!) Just watching her face as she looks at the castle, or hugs Tigger, or waves at the dolls in IASW, is absolutely priceless. I know that she won't necessarily "remember" these trips, but I always will. I say while you can take them at an age when riding the bus to the park is a novelty, there's no school schedule to worry about, and hey, they don't need a ticket, then do it!