Ask friends. Do they know someone who they think would fit you specifically?
Research them online. Many therapists in Austin have web sites, which are great sources of info. Mine (www.LeeEdwardsTherapy.com) has several pages' worth of info about how I approach therapy, what my background is, etc. Some therapists who don't have a web site DO have a fairly detailed profile posted at sites such as www.therapists.psychologytoday.com. Read the profiles and learn.
Call them. Once you think someone might be a good fit, call them up and ask if you can talk for just a few minutes about whether the two of you might be a good fit. You can say what issues you want help with; you can ask whether they have much experience with those issues; you can ask how they tend to approach those issues. If there are things you know you want in a therapist, I'd encourage you not to be shy about asking--gay/straight, liberal/conservative, advice-giving/quiet, whatever. I've had clients whose past experience led them to focus on my height or how my hands looked--safety issues. This leads us to:
Suit yourself. Give yourself permission to emphasize any characteristics you want. This is not the time to try to be broad-minded; it's time to find someone you feel comfortable with quickly and easily. The one exception is when you suspect that what'll be most helpful is picking someone who challenges you. I've had many women tell me directly that they're picking a male therapist because they feel unsafe with men and need to learn to have a healthy, safe relationship with a man. That can be a good choice. Most of the time, though, go with comfort.
You might want to check out my separate article about: What's the difference between psychologists and psychiatrists and therapists and....
Last thought about labels: generally speaking, you can interchange the labels "therapy," "psychotherapy" and "counseling." Whew. Still with me?
Please do some research because it's an important decision. Please please yourself. I hope you find someone you're very comfortable with!