I know there were 2 different factions, and the one backed by Sam Houston wanted to join the U.S., so why did it not happen until much later?
What was the support of the average Texan for joining the U.S. Or making its own country?
The Texans voted in 1836 to try to join the US. They called a popularly elected constitutional convention in 1845 which voted to accept the US Congress terms of joining the US (and to draw up a new State Constitution.
It seems that a majority of Texans was always in favor of joining the US.
The reason it took 9 years was that the US was very divided on whether they wanted Texas to join (or, at least, on what terms they wanted Texas to join).
US hesitance was driven by two factors:
Texas allowed slavery, so, the question of admitting Texas got caught up in the whole slavery vs anti-slavery debate, and the balance of power between the slave and free states.
Mexico had never recognized Texan Independence (indeed, had mounted two inconclusive invasions of Texas in 1842). Mexico made it clear that they would consider US annexation of Texas as an act of war. Many people in the US Congress had no desire for a war with Mexico.
Eventually, President Tyler, with some very cunning political maneuvering, got a Texas annexation plan through the Congress, and this was accomplished by his successor, President Polk.
Mexico did send troops into Southern Texas, where they fired on US troops, at which point, both nations declared war on each other. The US won the Mexican American War (perhaps somewhat more easily and comprehensively than they had imagined), and got a lot more land than just Texas (California, Arizona, New Mexico, Nevada, Utah).