Texas, being a small and essentially powerless republic on the fringes of what was considered the civilized world, had very few foreign relations. It was recognized by a small handful of European nations, mostly along the Atlantic-North Sea corridor due to Texan cotton being popular in European mills. Many nations were hesitant to recognize Texas, as they judged their relations with Mexico more vital.
Ultimately, there were only three foreign relations of particular import to the Texan Republic. In no particular order:
The reason the annexation required nine years of the "Republic of Texas" was the slavery debate, which was already consuming the USA. Neither major political party wanted to introduce another slaveholding state to the Republic and upset the delicate balance of power in the senate/electoral college. It was John Tyler, who assumed the Presidency following the death of President Harrison, who decided to begin the annexation process. Tyler was an avid believer in Manifest Destiny, and further, had grown resentful of the Whig party he'd been elected by. He believed annexation would make him popular enough to maintain power through the next election. He was wrong, and lost, but once he'd introduced a bill of annexation, the American public was enthused. Tyler's opponent, Polk, finished the annexation of Texas and laid the foundation for the Mexican-American war.
They did launch multiple attacks on Texas, however, in March and September of 1842. The first attack, led by Mexican general Rafael Vasquez, seized the city of Austin for a time, before withdrawing. Similarly, a French adventurer who served in the Mexican army named Adrian Woll led the September attack, which was essentially inconclusive. While the Mexicans could briefly capture cities such as San Antonio, they lacked the political power, popular support, or even requisite finances to truly occupy Texas.
As such, Mexico occupied a sort of boogieman's role in Texian politics. Defense against Mexican raids was a perennial topic in Texas politics, and was used in some of the republic's more bizarre political dramas. (See: A nearly-violent conflict over where the state archives should be stored.)
The Comanche absolutely cared about Texas, and vice versa. And in the worst possible way. Mexico hated Texas, sure, and launched a few minor campaigns in Texas. But the Comanche and Texas were true rivals and consistently in conflict. Early attempts at peace rarely lasted long, and ultimately Texas mounted an awkward invasiob of Comanche territory. The Comanche, in return, burned their way through the whole of the Republic during the Great Raid of 1840. We have few records of Comanche sentiment, but we know that many Texians violently despised the Comanche, and major politicians openly advocated for their complete extermination.
Once Sam Houston returned to the Presidency in 1841, the Comanche and Texas forged a new peace treaty, but it was borne less by a desire to co-exist and more by mutual exhaustion.
Oh, sources: Texans are particularly fond of their history. I've seen a lot of local historical society websites in my day, but Texas' is more impressive than most I've come across. I strongly recommend the Texas State Historical Society's online handbook, at https://www.tshaonline.org/home/
It covers most of this, with the exception of the European recognition, which you can find on the US State Department website. (For the record, I'm absolutely sure those are somewhere in the TSHA handbook, but that is the denser than a fruitcake and I didn't feel like looking for it.)