Because of the absence of political differences that would solidify into distinct parties, many historians have characterized the early years of Texas independence as "the pre-party" period. Opposition to the central government in Mexico before and after Texas independence unified the political leadership of the state, but did not create the pronounced differences that shaped the first party system in the post-independence United States.
However, the range of social and political influences that shaped the subsequent party system (as well as the state's political culture) emerged during the pre-party system, setting the stage for more than a century of politics. Populism, classical liberalism, and social conservatism all took root in the earliest days of Texas's political existence.
Populism was imported by Jacksonian Democrats who chased the frontier from Tennessee to Texas. Sam Houston, war hero and the new Texas republic's first regularly elected president, had the strongest ties to any political party in the United States. Houston had been Governor of Tennessee and reportedly was being groomed as the Democratic candidate for U.S. President by incumbent Andrew Jackson, himself an early populist.  Jacksonian populism was characterized by mistrust of excessive private power, particularly as exercised by large banks and corporations. In some ways populism continues the tradition of Jeffersonianism, which saw small landholders and businesspersons as the cornerstone of a healthy democracy. In the Jacksonian vision, the government could be used as a tool for restraining the accumulated power of large private interests. In the modern period this tradition of populism has been invoked by groups and individuals of diverse circumstances (rich and not-so-rich alike) to demonstrate their shared values with ordinary Texans.
Classical liberalism was planted by independent-minded Mexicans who had recently experienced intense struggles against the Catholic Church and the Spanish monarchy. These colonial institutions embodied classical conservatism, which upheld views of an unchanging "natural order" of society, in which people occupied fixed places in the social and political hierarchy. The Mexicans who fought against Santa Anna contributed a strongly liberal flavor to Texas political culture. One prominent Mexican, Lorenzo de Zavala served as the vice-president in the interim government immediately following the war for independence. Zavala had a long career of liberal activism before Texas independence.  He moved to Texas in 1829 under an empresario contract to settle 500 families there. He was appointed in 1833 by Santa Anna as minister to France. But he eventually came to the conclusion that Santa Anna would not respect the Mexican Constitution of 1824, and decided to resist as a Texan.
Social conservatism was conveyed by the many Southerners who migrated to Texas between independence and the Civil War. These Southern conservatives brought a distinctive blend of tradition, Christian social identity, racial hierarchy, and romantic individualism to Texas from the other states of what would become the Confederacy. The Confederate states, of course, lost the Civil War, but this particular brand of conservative political thought took root in Texas and would continue to influence culture and politics.
Several attempts were made to bring Texas into the union in the early 1840s, but these failed. Meanwhile, debate in the U.S. over allowing slavery in new states intensified as the presidential election of 1844 approached. Democrat James K. Polk, a protege of Andrew Jackson and strong advocate of Texas statehood, won the Presidency in 1844. With widespread popular support in Texas for joining the union, the Lone Star republic became a state in 1845. 
Between independence and statehood, the primary political issues were annexation by the United States (versus remaining independent) and expanding the western and southern geographic boundaries of the Republic (which had never been clearly specified). Vagueness in the definition of Texas's borders led to raids across the Rio Grande and territorial claims in what is now New Mexico. Geography and slavery combined to make Texas's relationship to the United States the dominant political issue of the period, as Texans with a strong attachment to the United States clashed with a succession of Texas nationalists, secessionists, and Confederates. For more on this period, see the Handbook of Texas Online entry for Unionism.
2 Alexander Hamilton, "Federalist No. 1" The Federalist Papers, pp. 101-114. link: http://thomas.loc.gov/home/histdox/fed_01.html
3 Ibid, p.74.
4 T.Neu, "Annexation," Handbook of Texas Online.