The hill country in central Texas is a treasure trove. City names like Marble Falls, and Granite Shoal are a big clue. Of course, there is the big site of Enchanted Rock, which is just outside Fredericksberg. There was a lot of seismic and volcanic activity here long, long ago. The largest documented blue topaz was found in Mason. There used to be a topaz hunting place there, but I don't know if it is still there. About 60 miles north of me (Austin), archeologists found a site dating back long before the Clovis people, which were thought to be the first ones here.
The trick is finding a place you can legally dig. Most of the land is part of private ranches, and ranchers take a dim view of trespassers. Of course, you are not allowed to take anything from a public park.
Vickie is right about the climate. The best time is in late fall and winter - after a freeze. The snakes will be hibernating, the bugs will be gone, and you won't sweat away half your liquids.