If you want reining lessons, go to a stable that is showing and doing well.
Any one teaching “western” but not competing probably doesn’t really know what competitive western riding is.
Don’t know which cities you are close to, but if you check with the NRHA, they have a list of trainers by state and you can find who in your state is a member and see what their web and fb pages tell you of what they are doing.
Cross training is great to be the best horseman you can be.
You can’t ever know enough when it comes to horses
You could also find out what reining association is local to you, see who their members are and how they do training and competing, maybe even find some reining shows to attend to see who you can make contact with.
Good luck!