Corpus, as we call it, will call for a slight deviation off of US 77 South. From Corpus, go back to US 77 [on I-37] and go down to Harlingen, where you'll pick up US 83 heading to McAllen. BTW, you'll be paralleling the UP's Brownsville Sub for quite a bit of the way on 77. If you have a scanner, monitor 160.74.[You might see one of the Rio Valley Switching Company's green and white units in Harlingen. To check that, exit in Harlingen on US Bus. 77, and follow it south about a mile. It runs right by the Harlingen Yard, and RVSC stuff will be on the north end of it.] Harlingen to McAllen is about 30 minutes, and your exit in McAllen will be Bus. 83, which follows the RVSC line through town. It's pretty dilapidated!
From McAllen to Laredo, it's the shortline Border Pacific RR, which terminates in Rio Grande City. You might catch a unit or two in RGC; they mimic UP's paint scheme. Between Rio Grande City and Laredo--nada! Do the posted 70 MPH where it's allowed, and you'll be up in Laredo in no time!
Your itinerary is ambitious for one day. Leave very early in the morning from Houston to accomplish it all in daylight!
Altamont Press puts out a Texas Timetable. You will find that very helpful.