The best starting point for gathering information about your MUD is TCEQ. All MUDs, whether created by petition or legislative action, are tracked by TCEQ.
If you already know which MUD you live in, you can start here and enter your district name (ex. Harris County MUD 459) to get to the district details page.
If you don't know which MUD you live in, start here to pull up TCEQ's map of water districts. Zoom in to where you live, click the sticky note button, select Municipal Utility District and click inside your district to pull up the district details page.
Once on the district details page you can click the "Documents" button in the top left to view summaries of the latest reports filed by your district (audits provide the most relevant information) or click "Run District Information Report" in the bottom right to find out who is in charge of the district (directors names, lawyers, engineers, auditor, etc). This will also tell you when the director's terms expire, be aware that registration for the election usually occurs months before the term expiration and if no one registers they can just cancel the election.
Law firms maintain all of the district's records and are required to supply the information upon request. Two of the most revealing documents about the MUD are the audits (see how much money is being spent on overhead) and bond applications (what is the developer getting above basic requirements).