One of the most interesting historic districts in Galveston is the "Old Strand" Historic District. This is where the people in early Galveston came to work and play. There are still some hotels here, along with trendy restaurants and decent bars. Lots of nice architecture and museums too.
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
The Tremont House is now a hotel run by Wyndham located in the heart of the Old Strand Historic District. The building was completed in 1880 by Eugene Heiner and expanded by famed Galveston architect Nicholas Clayton. It housed the main office of the Leon & H Blum Company the leading dry goods importer in Galveston. The company was owned by one of several Alsatian brothers and cousins who came to Texas in the early 1850s. The company also had offices in New York, Boston, and Paris, France, and served the Southwest U.S., the Indian Territories, and Mexico. The company was dissolved in 1895 and the building changed hands several times including housing the offices of the "Galveston Tribune". The building is listed on the National Register of Historic Places.