The Texas State Library and Archives Commission (TSLAC or Tea Slack) has you covered. 

TSLAC is a great staring place for researching county records, but it's hardly the only place to search.  Texas Fils has an online database of land records across the state. They also have probate records, searchable by each of the 254 counties. 

Despite Texas' notorious anti- federal leanings, it's worth it to check out the Library of Congress's resources for vital statistic and genealogical records that are grouped by state. 

If you're looking for court records, Research TX has its own online searchable database for court proceedings within the state. 

If you know the county you're searching in, Bexar county for example, you can directly search records within the county's own website.  Of course, the smaller and more rural the county, the more likely you'll need to visit in person. 

Want to visit a Texas County from the comfort of your own chair? Head to Verde County.