Start with a Birth Certificate (or Equivalent):
While other I.D. is allowable at the polls, those each require documentation such as a Texas Drivers License or Voter I.D. (Election Identification Certificate) to get.
The birth certificate is the starting point for native born Texans and other natural born U.S. citizens. Apply for a copy of your birth certificate with the Texas Bureau of Vital Statistics (or your birth state). You will need a drivers license number and an audit number from the drivers license to apply, but it can be the TDL number of a parent, sibling, or other close relative. The person ordering the birth certificate on your behalf affirms the city, county, and date of birth. Currently the fee to order a copy of your birth certificate is $22.
A birth certificate is now required before you can get a drivers license in Texas, so it is unlikely that you can enter your own unless your Texas Drivers License predated the current requirement. However, when renewing your drivers license in person, you will be required to show your birth certificate (or equivalent document) then.
If born in another state, you will need a certified original or copy of that state's (or U.S. territory's) birth certificate. Or a foreign birth document or other document confirming birth admissible in a court of law which establishes the voter’s identity can be used.
The Texas Drivers License and the Election Identification Certificate each costs $16 at this time.