Go across, buy drugs w/o a perscription, drink heavily, and get thoroughly sedated. You'll soon drive just like a local! Seriously, drug use by the local is a problem, and many gringos come here just to stock up on perscriptions.
The Trolley Tour leaves from the San Agustin Plaza - I think almost daily - and gives a great overview of Laredo history. If you are lucky enough to have Sam Johnson be your guide, you also get the perspective of someone who lived some of that history. If you are staying at La Posada Hotel (pricey but cool - considering it used to be Laredo High School back in the day), the Trolley Tour leaves from the plaza right in front & you have access to the Museum of the Republic of the Rio Grande next door. Trolley tours can be arranged for special groups but I'm not sure through whom - I can find out if you'd like.
Laredo (US side) is a very spacy city, with wide streets and detached houses. They told me that Laredo, compared to Nuevo Laredo, was more expanded while having the 1/3 of population. The houses had many architectural elements and decorations drawing inspiration from Mexican art. Here is a small example.
During my stay in Laredo in August, there was a national anniversary - like Texas Independence or something, I can't remember exactly. There was a celebration with an actual representation of some fights - don't miss it as I did! I have seen only some spots on the local TV :-(
If you do not already know the date, you have to ask about it!
Some evening, at the kiosk of the central square of Nuevo Laredo, I had the opportunity to listen to a free concert of a Mexican wind and percussion instrument band. If you hear about it, consider yourself lucky! Don't miss it!
Have a walk in the Park of Laredo (US side). I do not remember its name, but it is a beautiful relaxing place.
I remember I have met a guy there with a Mexican-like face, we didn't talk, but we were sitting on benches in some distance, and were just looking each other. He was alone, he came with his car, he put a fire on his barBQ and started grilling some meat. He had a cassette-player and he was listening continuously to the same song, a country-style one, refraining "...to cross the border !". I heared it over ten times and then I left...
Nuevo Laredo has its own arena, the PLAZA DEL TOROS. I missed it, because I was told that the arena is some kilometers away from the city, and I had not neither my own vehicle nor a company, that is almost necessary to share such an experience.