I am originally from Arizona as well! Tucson to be exact ...and I have sold many equestrian properties in both AZ and Texas and would love to help if you're still looking to buy, or have any specific questions?
There are currently some great deals on ranches in the Georgetown area that have both the mature shady landscaping, as well as the nice even terrain for riding. In fact, I know of two different horse properties coming on the market later this summer that should be priced near your basic price range.
One of the biggest concerns you'll want to be aware of, is the availability for water. Central Texas is currently in a sever drought and there are some areas of town that have had trouble with their wells.
Please don't hesitate to call or email if there's anything else I can do to help!
I would look south or east of Austin as much of the land to the west is too rocky and not good for horses. Otherwise you might consider buying a home close to a Boarding stable. For example for $250K you could get a nice home in Shady Hollow with about a half acre lot and board your horse just up the street at Woods Stables on Brodie Lane. If you want to see a list of homes for sale just let me know.
Your needs are best suited by a horse person REALTOR. There are different soil types here and areas not as suited for horses. East of IH 35, generally is blackland soil but you are on the plains where there aren't as many trees. West of IH 35 if the rockier Texas Hill Country. There are many good horse properties on both sides.
I understand, my sister lived in Phoenix and wanted grass and trees too.
My niece is a budding equestrian and she lives in Bastrop with my Brother and his wife.
Bastrop is a very up upwardly town as of recent with massive appreciation.
Please check it out!
Although I think any of us in Austin would love to help you (especially the brokerage I work for), but please take care to use a specialist so your investment isn't poorly placed. Good luck and WELCOME to Texas!!! :)