I went to Magic Kingdom and Animal Kingdom with my son when he was 2 years, 2 month old. We had a great time and he had a great time. He still remembers parts of it to this day (over a year and half later).
This is what I did .. this advice may or may not apply to your family:
1) Go early (like rope drop). You'll get more done before the heat and nap time
2) Nap in the park .. my son just napped in his stroller for about 90 minutes. He was out and was not bothered by the noises around him. .. we just found a quiet corner in the shade. If not, go back to your hotel.
3) Fast Pass anything and everything you can .. No line is better than a 10 minute line. After you finish your first 3 .. get the 4th, 5th, 6th .. as many as you can .. even for "kiddie" rides.
4) See characters .. but don't FORCE it if they are scared. HINT: .. stand between the character and your child while you hold them.
5) Character buffets (for dinner). Kids (that young) are free and you get a great meal and the characters come to YOU while your child sits in their high chair. It is a great way to see a lot of characters. I would suggest Crystal Palace.
6) Lower your expectations --- you will be moving slow with a child. Take joy in HIM enjoying all the cool stuff. You may not get on a lot of rides . but you will have a good time.
7) Stop and smell the roses -- get your photos taken, see parades, watch street shows.
8) If there is something he wants to do again .. do it.

. If they want to just play with things in the toy store for a while .. let them.
9) Don't stay late .. we made sure to leave before (or as) the night shows were starting. It was still past his bed time, but not much.
My son was entertained by the sights and sounds and didn't need anything else to entertain him throughout the day, so he was fine in lines, even longer ones. We had plenty of snacks and drinks on hand to keep up his energy.
He was actually WORSE when we went back when he turned 3 as he was a lot more independent and vocal then. Take advantage of the child being confined to a stroller.
While you can go on MANY rides at MK, I would suggest these for 2 year olds:
1) Dumbo/Aladdin
2) Jungle Cruise
3) Small World
4) Peter Pan
I would not suggest:
1) Haunted Mansion
2) Pirates
(they may be too scary)