If you're at all into wine, make a reservation at Pebble Rock Cellars. Some of the best Texas wines I've had (or wines anywhere for that matter) and every reservation gives you 1-on-1 time with the winemaker, who happens to be an Aggie too!
They don't really have a tasting room per se; if you make a reservation you're the only party there at that time. It's a tiny little building in Johnson City that he and his wife are running by themselves, and for a $25 tasting you get your money's worth.
His production amount is super small and all self financed. He was one of the winemakers at William Chris in the early 2010s and is great to talk to about the state of the TX wine industry. We just found out about him last weekend and already can't wait to go back.
Edit to Add:
Canyon of the Eagles on Lake Buchanan is (read: used to be, 15ish years ago) a pretty nice place to stay in the hill country. Not sure if anyone has been there more recently but it's definitely serene if you're looking for a nicer place to stay. Otherwise, you could just book an AirBnb for a night somewhere on 290 near Stonewall and hit a few wineries along the way.