Not sure if there is any not miss things to do but I celebrated my kids birthdays at DL at goofy's kitchen located at the Disneyland Resort. I called and ordered a cake from the bakery and it was brought out after we ate lunch. The characters came and took photos with everyone in our party. Goofy even came out and did a little song with all the kids in the restaurant. We had a lots of fun and lot of memories.
We do my sons birthdays there every year and it's always the same, a day at MK and either a character breakfast or lunch. Youngest DS just turned 2 last Saturday so we had the character breakfast at Ohana where he received his birthday button, an autographed birthday card, and a cupcake.
We took him to MK and rode his favorite rides. Everyone said happy birthday to him because of his button and then we let him pick out another sugary treat. He chose the Frozen cupcake at Contempo Café and made a huge mess.
I'm planning to take my oldest DS to Goofy's Candy Co. at downtown Disney for part of his 5th birthday trip.