Our last two trips have been with babies in tow (a 14 month old one trip and a 9 month old for another, oh and a one day trip with a 17 month old.) From those trips I can give these words of wisdom:
1. We loved getting to have one of the boys have their first haircut at the barber shop on main street and it is really reasonable (when you consider that it is disney) but make sure to make an appointment
2. The babies loved seeing the fab five (mickey, minnie, donald, goofy, pluto) and the friends from the 100 acre woods (pooh, piglet, Eeyore, tigger) so I would say your best bets for character dining are breakfast at one of the following: cape may cafe, ohana, tusker house or chef mickey's or dinner at garden grill and then any meal at crystal palace. My kids both disliked face characters when they were little like that. I guess stranger danger has set in by then so the princesses and alice and mary poppins did not go over well. For that reason you may not have a great time at cinderellas royal table, akershus or 1900 park fare. If your child watches a lot of disney jr then you could head over to hollywood and vine for lunch as well. Before booking all of those though- head to chuck e cheese and see how he reacts to the rat. He might like Mickey better but it would be horrible to have a ton of character meals planned and find out at the first one that he refuses to eat as long as one of those big headed animals is circling him like prey.
3. You can absolutely have your child sit between you and your s/o for any ride that does not have a ride restriction. For that reason the only rides you can't do are the mountains, kali, stitch, dinosaur, primeval whirl, soarin, mission space, test track, rockin roller coaster, tower of terror, star tours, barnstormer, and seven dwarfs mine train. I may have left one off but I can't place it right now.
4. At that age kids are definitely looking to you to know how to react to things. If you are nervous about how they are going to react they are very likely to react negatively because they see you scared. I always show the kids that things are fun and funny when we are at the park. As a result my kids are all sure that the alien on great movie ride has a bad cold, the ghosts on haunted mansion are running from their own farts (they're boys, what can I say!) or are playing the worlds best game of hide and seek together. This doesn't guarantee that they won't freak out (one time my oldest snuck a horror movie right before a trip and that made his little brother a nervous wreck every time the lights were low) but you have a much better chance of keeping them light hearted if you are very openly that way.
5. At that age it will be very important that your little one gets to let out some wiggles. It'll feel like the world's most expensive playground but really utilize that waiting area at dumbo, the playground under splash mountain, the boneyard, mission space's holding area, the exit to figment, etc. If you are the adventurous sort you can even add casey jrs splash pad.
6. If you aren't sure if your little one likes shows you should start with finding nemo at ak. While it isn't one you should get up to walk out of, it is the show that comes across as larger than life so if that doesn't hold attention then none of them will!
7. The train, the monorail, the tta, the ferry, the tram, the bus and the excursion train to rafiki's planet watch may well be the favorite "rides" of your little one.
Best of luck!