I'm not an expert, but I think the best exercise for kids this young is just being a kid! Active kids (i.e. those who spend most of their time playing and not watching tv) shouldn't need to have extra exercise programs because they spend all their time running, jumping, hopping, skipping, and climbing - all of which is great exercise for them. Follow a two year old around on the playground for an hour...you'll be huffing and puffing!
If they have the attention span for it, you could try some playground games like "Red Light Green Light" or "What Time is it Mr. Fox?" to get them running even more. Also there are lots of great gym programs like "Little Gym" and "Gymboree" if you don't feel like your little one has enough opportunity to play outside or on climbing structures.
But I think any kind of "adult exercise program" could be dangerous for their little joints, softer bones, and weaker muscles.
It is very important to provide and expose a 1-2 year old with a language rich environment. Children at this age need to be stimulated with words. Talk to your child a lot. As you take a walk outside show your child the trees and sky and point out the birds and children playing. Very soon your child will begin producing sounds and trying to talk with you. Ask your child what he/she sees? What they like. Stimulate your child with rich vocabulary from stories in books or just make believe stories as you spend time together. Encourage your child to try to express themselves verbally. Later at 3 yrs old begin noticing letters and sounds everywhere. Point to a stop sign and point out each letter and sound it makes. Ask your child if they can find a word that rhymes. Break words apart. In the word CAT, what is the first sound you hear? What is the last sound you hear? All of this creates phonemic awareness which later provides children with strong reading skills.
I would say hands on learning is at this stage. I remember my little girl wanting to pick everything up and it go straight in the mouth. This is a baby's way to learning. So surround her with all sorts of things she can touch, feel, and smell. You don't want to forget the visuals either. Colorful picture books and reading are a good choice too. You may want to create a special place in your home that has some learning stations. We had a basket full of baby books one side, a color wall on the other, a bucket full of bigger toys that had sound and movement (no small toys), an area where music could be listened to and some musical instruments (tamborine). So here you go! Your own little learning center right at your fingertips.
Interact with books.
Go places, and talk about what you're seeing there.
Read aloud to your child.
Allow opportunities for walking and movement to help develop gross motor skills.
Make reading books part of your morning or evening routine.
Allow opportunities to pick up and sort things of various sizes (but watch what goes in the mouth).
Remember, kids this age are developing cognitively, physically & emotionally so be mindful of opportunities to nurture each in your day.
In the book My Two Year Old by Ames & Illg there is a wonderful chapter on "The Mental Life of the Two Year Old or How He sees the World". You may want to check out this book at your local library and the corresponding one titled My One Year Old by the same authors to help you see the great potential developing at each age by your children. Most of all enjoy them!!!