UT is a behemoth college sports program. They play in the Big 12, and tend to be very good at the two NCAA sports that matter (football and basketball), with an emphasis on football. The last two years have been down years for them in football, but Mack Brown (head coach) is a great recruiter and, well, it's UT. They'll be good/great again, soon. The Red River Rivalry (football game/rivalry with Oklahoma) is a big deal.
Texas A&M is their top in-state rival, and also tends to be competitive (though not to the level of UT) in football and basketball. A&M played in the Big 12, but is switching to the SEC, thus (presumably) ending a long and storied rivalry with UT. A&M had great expectations entering the last football season, but blew a bunch of fourth-quarter leads and ended at 6-6. They fired their coach.
Otherwise, there's Texas Tech. They're "meh." And there's SMU, who is also lost in the shadows of the larger schools, but is moving from Conference USA to the Big East in the Big East's futile attempt to remain a viable conference.
NFL: there's the Dallas Cowboys; they're a big deal. Tony Romo is their quarterback, and is a very polarizing figure. He gets a lot of media attention, but generally has pretty inconsistent play. Jerry Jones is their owner, and tends to be pretty outspoken.
There's also the Houston Texans. They've sucked for much/all of their relatively short existence, but they're getting much better. They're a playoff team this year, but half their team is injured, so don't expect much.
NBA: The San Antonio Spurs were very good for a long time, but are showing their age. Their big star is Tim Duncan, who has the personality of a grapefruit.
There's the Houston Rockets, who are rebuilding. They had Yao Ming (giant Chinese man), but he retired.
There's the Dallas Mavericks, who are the defending NBA champs (they defeated the Miami Heat - with Lebron/Dwayne Wade/Bosh - in the finals). Their big star is forward Dirk Nowitzki. They just acquired Lamar Odom (who is married to one of those Kardashians) from the LA Lakers, after the Lakers botched (/the league conspired against them) an attempt to package Odom in a trade for another player (Chris Paul of the Hornets, who was just traded to the LA Clippers).
Otherwise, there's baseball. The Rangers are a very good team, and have appeared in the last two World Series (both of which they lost). Their manager is Ron Washington, who tends to speak his mind. And there's the Houston Astros, who suck/are rebuilding. They're moving from the National League to the American League.
High school football also tends to be huge in the state, and many top recruits come from Texas.
So there you go; that should give you a solid start, so at least you won't be clueless if the topic arises. Of course, this is assuming they want to talk about Texas sports.
Anyway, HTH. It certaintly helped me procrastinate.