Day trip suggestions are very welcome.
What time of year will you be in San Antonio?
The trips you might be interested in will depend on where you are from and where you have been before, probably, but I will list some options:
1. Fredericksburg is a small German-heritage town northwest of San Antonio (about an hour's drive) that is a popular destination for visiting shops and restaurants. There is also the Museum of the War of the Pacific there (don't know exact new name - it used to be called the Nimitz Museum).
2. In association with a trip to Fredericksburg or separate, there are numerous wineries in the general area of Fredericksburg.
3. My personal vote as the most beautiful drive in the eastern 2/3 of Texas is on 337 between Medina and Vanderpool, west of San Antonio.
4. Gruene is a small town just on the north edge of San Antonio that has lots of little shops, some restaurants, and Gruene Hall, the oldest dance hall in Texas. On Sundays, there is free music all day until the admission-required evening shows start (which are also great!).
5. Austin is to the north about an hour or 1.5 hr. It is not as tourist-oriented as San Antonio, but the capital building/tour and grounds are interesting, as is the Bullock Museum of Texas History. The Henry Ransom Center (free) is small, but has rotating exhibits from its collections. The Elisabet Ney Museum (scupture) is a small, often overlooked museum that is pretty interesting. There are lots of restaurants downtown, but not so many shops. For quirky shops, go to South Congress, just across the river from downtown. There are plentiful restaurants there, also.
6. If you like barbecue, there are many small towns in central Texas that have great barbecue. Between San Antonio and Austin is Lockhart, which has both Black's and Kreutz. Both are good (although I prefer Kreutz!).
7. A little farther afield, Corpus Christi is about 2 hours south, on the coast. It has the U.S.S. Lexington aircraft carrier, which has a lot of historical information and is interesting to see, as you can visit much of the ship. There is a nice waterfront with some decent restaurants, etc. Not much shopping.