Although mostly a personal subscription database aimed at genealogists, Fold3.com has a number of free sources, digitized and (to a limited extent) indexed versions of National Archives microfilm. At this point (June 2015) they include:
At this point Civil War military records, both federal and Confederate, are behind the pay wall, as are federal pensions and many other records. However, Fold3 often runs 7 day free trials, or you can pay $7.95 per month, or $79.95 per year, which is a great deal if you will be needing to use these records over time. There are often special collection offers, such as all Civil War records were free for two weeks in April 2014, commemorating the month the Civil War began. It's always worth checking.
The Robert R. Muntz Library has a small State Depository for state documents housed on the 3rd floor of the library. As you come off the stairs, turn right and go down to the 7th row on your right. It falls between the regular sized books and the oversize books. The collection includes General and Special Laws, Senate Journals, House Journals, and archeological reports, among other documents. All are listed in the catalog. In addition, the Muntz Library has the Texas General and Special Laws for the 19th century on microfiche in the reference area.
Summaries of each census since 1790 are included in the series America in Two Centuries, an Inventory, with a volume for each census. The years 1790, 1800, 1830, 1860, 1870, 1880, 1890, 1900, 1920, and 1930 are at HA 201 (regular sized books). The years 1810, 1820, 1840, 1850, 1910, 1940, 1950, and 1960 are at HA 201 (oversized books).