Where should I go to fix up a friendly stray cat?

Hi! I’m an 18F college student who has no experience with pets. Last week, a stray cat appeared in our neighborhood out of seemingly nowhere and is very, very clingy and friendly. She will chase us down everytime my family steps out of the house and tries to come inside. She’ll roll around and sit on our front porch for hours. I have a feeling she might have been abandoned, as we’ve seen her live on our front porch, in the bushes, or in the gutters, day and night. She is so extremely friendly and vocal, but there is another stray cat that’s rather territorial and keeps bothering her (this territorial one seems to be content with his outdoor life). How should I go about helping this baby out? I feel so bad. My parents are very paranoid about disease as she is quite dirty, so how do I go about getting her washed up and checked for disease? Is there a place in Houston (we live near Sugar Land) that is able to spay, check her vaccines, etc on a budget?