What may be happening is this. Let me know if this isn't it. You may possibly be thinking the sign means:
Drive friendly. Do so by driving specifically the Texas way, not another state's way.
However, what it means is this:
Drive friendly. Driving friendly is the Texas way.
It just means don't drive crazy in general. Don't go twice as fast as the speed limit. Don't harass people. Don't drive drunk. Don't do bad stuff in general.
As far as the driving rules go, Texas is usually pretty normal. Their driving rules are about the same you see in any other state. There might be a few states with really weird rules that matter for normal, everyday driving, but Texas isn't one of them.
If you drove from California through Arizona and New Mexico to Texas, then just drive the same way you do in Arizona and New Mexico. For normal, everyday, passenger car driving, the rules are basically the same between AZ, NM, and TX. (I think CA is similar as well, but I'm not completely sure. I've lived in AZ and TX though, and they are very similar.)
By the way, for English in general, the grammar and such they use in those signs can have either of those two meanings mentioned above, but the cultural context says it's the second one in this case. I'm guessing it's the same in the UK, Canada, and Australia, but there may be one or more English-speaking countries where the culture is different in this matter.
Something this reminded me of just now: This isn't really a cultural context issue, but a word definition issue. In New Zealand, you might see this type of road sign:
Many Americans won't understand what that sign means. In our form of English, "zip" is basically just a verb, and a lot of times, schools don't teach this specific dialectual difference. Some Americans may figure it out immediately, but many of us would probably at least have to think about it for several seconds (particularly if we can't see the road very much further ahead, because of hills and stuff blocking our view).
However, if we translate that into AmE, it basically becomes:
For New Zealand, "zip" is easily both a noun and a verb, but for the US, it's mainly a verb. The sign is talking about two roads merging into one, and it's saying to have cars from both roads take turns, one car at a time.