Hi betenoire!
Glad to hear you and Nick will be staying downtown because you'll be in the center of everything that is important and easily accessible.
Shopping - well, there are smaller shops around, but the Rivercenter Mall is right there as well. Not really my thing either, but it's massive and you can spend quite a few hours in there.
The San Antonio Riverwalk - this is a high recommendation. Here is a map where you can see everything - restaurants, clubs, shopping, the museum district, etc.
http://www.thesanantonioriverwalk.com/
I honestly am not too familiar with the taxis, however I would think that the hotel would be able to provide you with some.
Food - It will be everywhere, but I highly recommend Mexican! Bourdo's Bistro, County Line (for Bar-B-Q!), and Iron Cactus are a few good places I have been. There's also the usual touristy spots like Hard Rock and Rainforest Cafe, but I recommend anything but those.
Just remember too, it's going to be hot as hell here in July. Be comfortable, and I hope you guys enjoy your stay.
Here's the SA chamber of commerce site also -
www.sachamber.org
And see the Alamo.
Kellen