@Ardauto
I lived in Texas for 14 years until 2006. Texas is the second largest state in the US in population and geographic area. Despite popular myths about cowboys and rodeos, it is primarily an urban state. It has two major population centres: Dallas-Fort Worth and Houston. Austin is the capital and hub for politics, academia, and high-tech; San Antonio is a major hub for tourism and the military. The first three are metropolitan and diverse. Dallas-FW is probably the leader in terms of cultural activities and attractions such as museums, although Houston is a close second. Both are large, dense, sprawling urban centres. Austin is attractive for its status as the Texas oasis of liberalism in the sea of red (i.e., red state politics, or republicanism). San Antonio is comparatively laid back and somewhat more traditional, due to its large middle-class Hispanic population, and very large number of active military families and retirees. For competitive athletics, all four would have something to offer. I would avoid the other cities unless there is some specific, intrinsic value located there, such as a sponsoring family with which you have close ties. However, the four areas listed above could be overwhelming for high-school students to navigate and survive in on their own. Assuming you have secured the necessary visas, etc., you should find a school which will accept your children and which specialises and excels in the academic or athletic activity they are most interested in pursuing. For example, you could search for "Texas ju-jitsu high school rankings" to find possible candidates. There are many private schools in Texas, although a significant number are affiliated with a particular religion (e.g., southern Baptist) and have a specific agenda for indoctrination. If you are Catholic, the large Hispanic populations in San Antonio and El Paso (although not recommended for other reasons) may be more relatable and welcoming.
Regarding the culture and accent, the Texas twang/drawl/accent isnt hard to acclimate to, and certainly no worse than Kiwi ganglish or Canadian blandness. Most Texans will speak American English with fewer idioms or language anachronisms than you would find in (much) smaller countries.
The best things about America - and Texas - are the deeply ingrained beliefs in equality and possibility; and indefatigable faith in progress, technology and the future.
Note: I combined the cities of Dallas-Fort Worth, which are ranked #3 and #5 respectively, in terms of population. The Dallas-FW metroplex also includes Arlington. Combined, Dallas-FW (as they are often lumped together) are larger than San Antonio, which is second largest based on the administrative limits, but smaller than Dallas-FW, with or without Arlington.