I have turned mesquite, and I find that it turns reasonably well although it can be a bit brittle. Finishes nicely.
One caution about using locally-grown wood - its probably going to be 'green' when you get it. It will turn beautifully, but you may be disappointed to find checks and distortion in those projects six months later. The rule of thumb is that it takes one year per inch of thickness for timber to dry out (ie, for the internal moisture content to equalize with ambient humidity).
If you look around, you might find sources of dry turning stock, especially for smaller projects. Shops that do custom furniture and cabinetry often have a bin of hardwood offcuts that you can buy for a very reasonable price. The only thing to be careful of is grain direction - if the offcuts are too short, the grain won't be in the right direction to allow them to be used for spindles. It's still possible to turn them, but turning narrow cross-grain stock takes a bit more skill than you might have just starting out.
When I first started out, I discovered a factory near here that makes treenware - bowls, platters and other kitchen items made of wood - and that sold odd scraps as 'kindling'. I was able to pick up quite a bit of maple, ash and birch from them - until they discovered that a better business decision for them was to save it to burn in the winter to heat their shop.
Another option is pallet wood. Yeah - throwaway wood, but sometimes you find pallets that were used to ship items from overseas and that are made of (by US standards) pretty exotic wood.
Home Despot and similar big box stores do carry a limited supply of a few hardwoods - typically poplar, red oak and maple. And there could be supplier who specializes in hard woods - but don't expect it to be inexpensive!
