There are three major things to be done at the line of century old buildings named the Strand.
The shops make up the first floor of many of the buildings. You can find anything here but my favorite would be the craft-type materials. Antiques are abundant as are beach wear.
There is the complete spectrum around the Strand. There's casual sidewalk cafes all the way up to elegant gourmet dining. I usually enjoy sitting outdoors and watching the world go by with a cool drink. But be mindful that in Texas during the summer it may be too hot to sit outside with a cool drink.
3) See the historic buildings
The buildings are really amazing. On every block are historic landmark plaques, most of them State rather than National.
The Strand is also close to many of the museums around Galveston such as the Railroad Museum, Elissa and the Pier 21 Museum, Mardi Gras museum, trolley cards touring more of the historic homes, and more
Check out a map at http://www.thestrand.com/maps/maps.php