Twitter can serve as an excellent tool to stay current on topics that interest you. 
This list provides information on various organizations that use Twitter to share information on Texas legislative developments.  The list is not comprehensive, but will serve as a starting point to help you discover other Twitter sites on this topic.   
Texas Legislative Reference Library
We tweet about the legislative process and share reports and news on issues of legislative interest.
The Texas Legislature
Up-to-the-moment information about Texas bills and committee meetings.
Texas Legislators list
We've collected the feeds of Texas legislators who use Twitter into one list, making it easy to follow multiple individuals in one location. 
State agencies with a social media presence -
State agencies may post about legislative developments during session.  Use this link to see which state agencies have Twitter feeds.  
News and journals:
Many regional newspapers use Twitter to share information about Texas political developments.  Here are feeds by some of the larger Texas newspapers:
Fort Worth Star Telegram:!/txpolitics
You will also find feeds specializing in regional politics:
mySA Politics (San Antonio)!/mySA_Politics
Texas Watchdog (Houston politics):!/texaswatchdog
Several journals specializing in Texas politics and government use Twitter to post updates about news and articles:
LegeLand (Austin Chronicle):
Texas Public Policy Foundation:
And finally….
Multi-state trends
If you are interested in tracking developments in state legislatures across the U.S., the National Conference of State Legislatures (NCSL) is a good resource.  NCSL is a bipartisan organization that reports on legislative news and developments in all 50 states. 
National Conference of State Legislatures