Cheaper ways to travel in and around Katy?

Ok so my psychiatrist moved from Memorial to Katy and for private reason I have to see them in person and I don't want to start all over with a new doctor right now. I don't drive so I've been relying on Lyft and Uber to take me to and from there.

Anyway, there has to be a cheaper option to get to Katy from Westchase besides spending 24-30$ TWICE on a lyft, or going all the way to downtown just to take a bus to Katy.

Also since I'm rarely over there, what if I want to go around Katy like go to Katy Mills, Asia Town, or the HCC at Katy? What if I just want to visit a relative thats 15 miniutes away in Katy. Is there another way to travel within the city without shillings so much money to rideshares?