I did start my kids on board games about this age, but you will need to be extremely patient! :) My daughter handled it better than my son.
I agree with Candy Land. I'm not sure if they still make it but there used to be this Candy Land castle game (not the original one) that had shapes that came out of a chute and they placed it on a mat. It was fun, very easy and perfect for 2-3 y/os.
Zingo is also a good suggestion.
We played a lot of Go Fish at that age too. There are lots of different card games out there but we like the eBoo brand. They have cool graphics and very sturdy cards.
Memory is good, and yes you can always modify it by only using some of the cards. My son LOVED memory and even at 5.5 still wants to play it all the time. We have various ones to mix it up and it's always been a favorite with my kids. eBoo also has great memory games too!
Hi Ho Cherry-O and Sneaky, Snacky Squirrel are also good. Hungry Hippos is fun and helps with fine motor skills. I would avoid Chutes and Ladders. My kids had a hard time with it at 2-3. The way they number the squares is confusing, IMO, and it made me frustrated. Choose other games for counting.
Games have come a long way from when we were kids! Just look through the game aisle at Target and you'll fall in love with quite a few, I'm sure! Just look on the box for ones that don't require reading and I have found that for the most part, the age range suggestions on the games are pretty accurate.