Itsy Bitsy (or Eensy Weensy) Spider is probably the best known. Like this:
The itsy bitsy spider climbed up the water spout [thumb to little finger, twist hands to make the spider climb]
Down came the rain [palms forward, wiggle fingers and lower hands to make the rain]
And washed the spider out [hands together in front of face, swish down to the sides]
Out came the sun and dried up all the rain [touch fingers over your head with arms in a circle]
So the itsy bitsy spider climbed up the spout again [spider motion again]
What other songs have motions that go with them? There’s “The Farmer in the Dell” which I seem to remember has something, but I don’t remember what. And “The Old Brass Wagon”:
Which I’m sure must have something.
What others are there? (I have daughters 2 and 3, in case you couldn’t tell.)
Let’s not forget the classic “I’m a little tea pot”:
I’m a little tea pot, short and stout (both hands on top of head)
here is my handle (one hand on hip)
Here is my spout (other hand held out to side, bent at elbow to resemble handle)
When I get all steamed up , hear me shout
Ti-p me over (tip toward the “spout side”)
and pour me out!
Darndest thing…I don’t even have kids… 
I choreographed a group of 5 year old girls for “Under the Sea.” It was pretty easy.
Well, there’s “The Three-Cornered Hat”
My (point to self) hat (point to top of head), it has three (hold up three fingers) corners (something, but I don’t remember what)
Three corners has my hat (repeat gestures as appropriate)
And had it not three corners (again with the gestures)
It would not be my hat! (a bit more of the gesturing)
Then, you repeat, only leave out one of the gestured words each time (my, hat, three, corners) but retain the gestures. With each repetition of the song, leave out another word.
Hilarity ensues.
There’s also the classic “Head, Shoulders, Knees and Toes” with the pointing at each.
That’s all I can think of off-hand.
“Do Your Ears Hang Low” (the cleaned-up version of “Do Your Balls Hang Low”) is now a pretty well-established kids’ song, and it has standard gestures
Do your ears hang low (fingers together, thumbs out and touching sides of head; “ears” then flop fingers-downwards)
Do they wobble to and fro (wiggle fingers)
Can you tie them in a knot, can you tie them in a bow (pantomime tying a knot)
Can you throw them over your shoulder (pantomime throwing a sack over the shoulder)
like a Continental soldier (salute)
Do your ears hang low? (repeat first gesture)
There’s also Little Bunny Foo-Foo, but all I remember of that is the Good Fairy part: You touch your thumbs and index fingers together, spread your fingers, and put your thumb-finger loops over your eyes, with palms toward your temples, to form a sort of mask.
Then you have things like “London Bridge” and “Ring Around the Rosie”, which have a sort of game associated, involving many children.
And of course, there’s always the Hokey Pokey, which is what it’s all about.
Gray Squirrel – a selection people beg me to not favor them with.
Put your hands up in front of you in a squirrellike pose, and … hit it!
Gray squirrel, gray squirrel [rock forward and backward]
Swish your bushy tail [wiggle ass]
Gray squirrel, gray squirrl [see above]
Swish your bushy tail [see above]
Wrinkle up your funny nose [surely you can guess]
Put a nut between your toes [see above]
Gray squirrel, gray squirrel
Swish your bushy tail.
Come to the next Philly Doper Dinner and see me do it in person!
If you wanna hang out you’ve got to take her out; cocaine. turn pockets out
If you wanna get down, down on the ground; cocaine. squat down
She don’t lie, she don’t lie, she don’t lie; cocaine. shake finger and head
If you got bad news, you wanna kick them blues; cocaine. kick football
When your day is done and you wanna run; cocaine. run on the spot
She don’t lie, she don’t lie, she don’t lie; cocaine. shake finger and head
If your thing is gone and you wanna ride on; cocaine. horse riding pose
Don’t forget this fact, you can’t get it back; cocaine. hand to head puzzled
She don’t lie, she don’t lie, she don’t lie; cocaine. shake finger and head
She don’t lie, she don’t lie, she don’t lie; cocaine. shake finger and head and big loud snort