We love Galveston! There are two great places to stay depending on what you want to do and how much you want to pay!
Moody Gardens Hotel is a great! They have a really nice pool area plus your stay includes admission to their own water type park (big pool with sand, splash zone) within walking distance. It is also in walking distance to the Moody Gardens Triangles...an aquarium, a rain forest, and a 3-D/Imax type theater. They have a restaurant to eat at and are extremely family friendly. Moody Gardens is in the heart of Galveston and close to other restaurants and attractions. It is in walking distance also to Schlitterbahn Water Park.
The other place is 20 miles west of Galveston called Pointe West. You have to drive into Galveston, go to the sea wall and then drive the 20 miles. It is only 2 or 3 years old and offers 2, 3 and 4 bedroom cottages. They have 2 pools...one beach side and one bay side. Offers toys, cabanas, kids activities, etc. We enjoy staying here also because it allows us to cook our own food and the two bedroom condo was big enough for the four of us (2 adults, 4 1/2 year old girl and 2 year old boy). You have to go grocery shopping however before you get to the property b/c it is so far back into Galveston to get to the regular grocery stores.
http://www.terramesahospitality.com/pointewest/Home.do
We have stayed at both places. And enjoy every time we go. Galveston is really revitalizing itself and bringing in new sand. The great thing about Pointe West is that it has private beach area and walkovers. Whereas Moody Gardens, you will have to drive to a beach.
There is a train museum in Galveston that our kids really enjoy as well as just walking along the beach and chasing the birds and getting all sandy!!! Also depending on how you go (driving or flying) in Kemah, there is a fun boardwalk area to go to with a carousal and rides and there is also NASA.
Lots to do and see! Hope this helps!