We've been taking our son starting when he was 7 1/2 months, and now he is almost three (so these are my tried and true pointers).
-Take a comfortable stroller that reclines. My son has always been able to sleep in his, and for us it was much easier letting him nap in the parks then try to go back to the hotel. When he naps, we go on the height restricted rides and use child swap.
-For a bag, we've always just used a backpack. Filled it with diapers, wipes, hand wipes, lysol wipes, autograph book/sharpie, sunglasses, snacks, sippy cup with water, hat, extra change of clothes and glow sticks/light toys for night.
-Child Care Centers are awesome! If you are close to one and need to change a diaper (or have a diaper that needs an extra set of hands), they all have wonderfully clean rooms with actual changing tables. It is so much nicer than trying to do it in one of the rest rooms.
-Schedule some buffets/all you care to enjoy meals (especially character meals!), b/c you won't be charged for your toddler. Our favorites are Tusker House and Garden Grill, which is also easier b/c the food is brought to your table family style.
-Think about your FastPass selections and choose the busier rides that you think your toddler would like for FastPasses. Lines with toddlers are HARD! Snacks and little toys help in line, but I don't think I would get in anything over 20 minutes with mine. During early afternoon when the lines seem to get the longest, the carrousel and Mickey's PhilharMagic are good, shorter wait options.
-Make sure you give your child some time to run around. Between strollers and lines, there really isn't a change to run around, but there are some great spots. MK - playground inside Dumbo or the little playground under the railroad track by Splash Mountain; EP - lots of spots here: interactive play area after Journey into Imagination (can also enter through exit), the aquariums w/Bruce's Sub House after the Seas with Nemo (can also enter through the exit), the car display area after Test Track (again - can also enter through exit, can take turns riding single rider, and it's a nice, big area for napping); AK - the Boneyard! Great playground area. Also, make sure you take the Wildlife Express Train b/c 1) it's a train, and 2) they have the world's cleanest petting zoo with goats that don't mind toddlers. DHS - this one is hard. We never spend a whole day there with our toddler.
-Here are some other things we discovered: Our toddler HATES fireworks, so we usually skip. Shows also do not work well for us, b/c toddlers sitting that long is a challenge. Be prepared for them to change their mind about what they want to do - and just go with it. There's plenty to do (once we arrived at park open and headed for Pirates with no line. Well, as we started to head in, he started screaming he didn't want to go on Pirates - he wanted to go to the Tiki Room. Yep, we spent prime morning time in the Tiki Room...)
-And here are some fun ideas for a toddler's first trip: Get a Mickey Ears Hat with their name on it at the Chapeau on Main Street as soon as you arrive. They are adorable in photos, come in toddler sizes, and ours has lasted 2 years and a whole lot of trips. If they haven't gotten a first haircut (or even if they did), get the first haircut package at the Harmony Barber Shop. They only offer a very limited number of reservations, but we went at parking opening without one and got in pretty quickly.
-There's also a great old Disney Sing-A-Long video called
Disneyland fun that shows the actual characters. Yes, it's dated - and the wrong park - but it's a great way to get them exposed to what the characters look like and could make them more comfortable at character meets.
And get them a Mickey Ice Cream Bar while you wait for the parade!