I've done this before, but as we approach Texas' 84th legislative session in January, Grits thought it prudent to iterate a few handy sources for those following the action from outside the pink dome.
Capitol website: Search by bill number or perform word searches. (Be sure it's set to the appropriate session.) Or review all current Texas statutes online. Especially useful for bill research are the reports sections, both the general, pre-defined reports and narrower ones you specify. The capitol site also provides a password-protected, personalized bill tracking service and bill/meeting alerts. These are the equal of professional services that cost thousands of dollars and have improved every session. For example, this session for the first time, pdf versions of bills on the capitol website include hyperlinks to the underlying statute they're changing. (The Legislative Reference Library has a tool that previously assisted with that and will still be needed for paper versions.) The only things the paid services have that the capitol site doesn't are blow-by-blow updates from the floor, especially during the amendment process, and more detailed minutes/notes from committee meetings that give you a time-stamp for witnesses' testimony so you can go back and find it amidst the countless hours of video on the capitol website. Otherwise, for all but full-time professional lobbyists, these free services should more than meet your needs. The House and Senate websites are operated independently and contain unique information about individual members and committee activity in those chambers including streaming and archived video of committee hearings.
Legislative Reference Library: The Legislative Reference Library on the second floor of the capitol is an underutilized state gem that few people outside legislative staff, the better lobbyists, and a few appellate attorneys truly appreciate. The staff is friendly, helpful and patient. Great for researching legislative history and intent - its main function - but their whole collection uniquely focuses on Texas state government and I've learned a lot over the years just combing the stacks and reading. Here's a brief list of their specialized collections. Their website is full of great current and historical information and Texas-specific tools and databases. They've also got a nice (if regrettably blog-free) statewide list of Texas media outlets. And perhaps most usefully on a day to day basis, if you've ever been in a legislator's office and seen a stapled, legal-sized compendium of daily news clippings floating around staffers' desks, these are the folks who compile them. Happily, these days they also put the daily links online.
Texas Tribune directory: Most folks are aware of the Texas Tribune's nonprofit news stories, but their state official directory lists contact info and staffers for every member. One of the best, comprehensive, free sources of that information and it seems to stay pretty up to date.
Legislative Budget Board: Provides access to agencies' appropriations requests and tracks the progress of House and Senate budget bills. On the criminal justice front, they also track adult and juvenile prison population trends and numerous other quantifiable policy measures.
If you've got those tools, a telephone, email, and a Google News feed, and if you don't mind spending hours watching hearings online, you've got access to most of the methods available to professional journalists and lobbyists who track this stuff for a living.