Texas state records usually date to at least 1903 though older records can be found if you start looking at individual county offices. But for the main state database, that's the date they start. Texas genealogy usually starts with the standard vital records of birth, death and marriages.
Texas Vital Records
There are some privacy restrictions on vital records in Texas, but the specifics will vary depending on what type of record you are looking for. If you are looking for records of someone you are a direct relative of, then you will not have any difficulties at all. And by direct relative, they mean children, parents, siblings, spouses or grandparents. For more distant relatives or for people not related at all, then the age of the record will matter. Birth certificates will be issued after 75 years, and death records can be ordered after only 25 years in that case.
To order a copy of either a birth or death record, for Texas genealogy purposes or anything else, you need to get the proper forms from the Department of State Health Services website. You can either mail it or take it in person to the Vital Statistics Office in Austin. The fee for either record is currently at $22 (even if they don't find the documents you want).
Marriage records are different and less straight-forward. The Vital Statistics Office can issue verifications of marriages ($20 each) but you can only get copies of actual marriage certificates from the county offices where the marriage happened. The state collection of records for this only goes back to 1966, so anything older will have to come from the county clerk's office anyway. You can also make requests at the county clerk's office for birth and death records that are older than 1903 though records may be incomplete.
The State Archives
There is more to genealogy research than just vital records. The State Archives can provide a number of other avenues for information, and they are typically free to the public if you can visit their research room in person. There are no pre-1903 vital records in their collection but there are many other historical documents that can provide more Texas genealogy data for you.
The Texas State Library and Archives Commission is located in Austin, and they are open to the public for standard weekday business hours. Much of their collection in on microfilm so you really have to visit in person to do your research. There are passport applications, newspaper archives, tax rolls, pension applications and voter's registration collections.
Of course, you don't have to research Texas genealogy all on your own. By joining a good historical society you can have access to their own collections of material and just get some assistance from other members. There are dozens of groups all through the state, many focusing on individual counties or regions in Texas. The Texas State Genealogy Society and the Texas State Historical Association are two good ones to start with that cover the whole state.