I don't claim to be any sort of expert, but I am a certified Early Childhood SPED teacher (although I teach mostly upper elementary grades) and I read a lot. So like I said, not making any sort of miracle-worker claims, but I'm going to share some activities that you might like that work for my child.
La Clase de Espanol: I don't speak Spanish. I have a half-way decent receptive language and poor expressive language, but we really hope to help Ellie obtain Spanish proficiency. Since Ellie has loved her See and Learn flashcards, I just downloaded some books in Spanish for Matt to read with her. Check out DSFOC. You have to sign up, but they don't spam you and it's free. Seriously. Go download some books (in English or Spanish) and some number cards, and some parent tips. We are just starting the Spanish stuff, but she loves the English versions.
Counting Books: There's a few good ones, also at DSFOC. And we've been counting everything. And signing number songs.
Letters: Aside from reading Chica Chica Boom Boom over and over and over, Ellie has been playing with magnet letters. Tip for when you don't want your kid hanging around the kitchen? Put the letters on a cookie sheet in a room that doesn't have stairs going down to the basement, a hot oven, or knives!
Matching: This is the easiest one to set up. Make a list of names in your family. "Ellie" "Mommy" "Daddy." Write the names each on a different color flashcard, make two copies of each. Match away to work on colors and name recognition at the same time. Ellie does this while I'm at work.
Fine Motor Skills: We bought Ellie a cheap RoseArt easel for Christmas, and Matt set it up with the legs upside down, so it's as low as possible. Since it's pretty light, we put clear plastic boxes holding Ellie's LEGOs, blocks, and crayons on the tray to weigh the easel down and provide our little "I don't wanna" walker with support. She loves to do art, and will announce that her work is "pwetty!" I normally dislike RoseArt stuff, but in this case, it's the shortest, so it's the best for Ellie. The bonus is that the easel forces her to stand, and she likes it!
Videos: Ellie loves herself. So I take videos of her talking, and she watches them on my phone. I have no idea what this means developmentally or if she's just kinda vain, but nothing gets Ellie talking like her own image onscreen.
Most Important... We talk a LOT: I believe Ellie is a talker because her mama is a talker. We keep up a pretty non-stop conversation describing everything, and really praise every new word.
I honestly believe different kids have different activities they love and different gifts because of their loves. I believe I have given Ellie every opportunity to be an early walker, and she's nearly two and up to about five steps. I know that the walking will come, and I'm glad she's a talker and a bookworm. I hope you find these tips helpful, and if there's any great activities I've left off, please leave them in the comments section!
And yes, this post addresses the particular learning styles common to kids with Down syndrome, but these work for all kids. And if you have a kid without Down syndrome who just happens to be really short, the RoseArt easel trick is awesome.
P.S. I pin a lot of the activities I find here.