I second (third, forth, whatever) the harness idea. My kids haven't needed one (yet...), but I see lots of parents with them. If you have a "sqirmy-no-stroller" kid or a "runner" then they can save the trip.
No one has mentioned characters yet, so I'll add my 2 cents of advice from taking my kids at about the same age:
1. Start slow. Try to start by watching a parade or show so the child can see the characters from a safe distance. If this freaks her or him out, do not proceed to step 2. There will always be another day or another trip to try again.
2. At greeting areas:
-Make sure child is in a good mood and not hungry, thirsty, or tired before joining queue.
- pick up the child and hold her (or him) as you walk up to the character.
- Say (loudly), "Hi Mickey (Tigger, Pooh, etc.)! This is (child's name) and this is her first time meeting a character." Character performers can be really good at drawing a shy child out or being low key to not overwhelm a child who is new to the experience IF they know that they need to.
- Keep holding the child, and keep yourself between the child and the character when you pose for photos. (Though you can also give the child a chance to touch the character if she or he wants to shake hands or pat Mickey's nose.)
- Even if things are going well, don't press your luck and try too much at once. Save it for next time and you'll be more likely to get a next time (instead of a character phobia).
In general, travel with a toddler will go better if you take special care to respect the child's personality and her (or his) needs. Take everything at a slower pace (split up the adults in the party if you need to), take lots of breaks, stay well watered, fed and rested, and enjoy the moment. If the toddler is having fun playing with a few pieces of mulch from a planter, don't feel like you're wasting time if you don't rush off and get in line for a ride instead. Seeing Disney through a toddler's eyes can be truly magical. The rides will always be there (well, most of them anyway), but the toddler will never be this age again.