This is a great topic to discuss. I'm not sure that texting does vary depending on one's audience, but it should! One thing you might want to do is interview some people of different ages and with different relationships, a kind of survey to see what some difference are (or are not.)
What I have noticed is that the younger a person is, the more likely he or she is to use various abbreviations, shortcuts, and initials. For example, I see "u" for "you," and "BTW" for "by the way. Older people don't seem to do that quite as much.
I have also noticed that younger people do not seem to make adjustments for whom they are texting, for example texting a teacher or an older relative. The use of abbreviations, shortcuts, and initials does not change at all. This can create a communication barrier for some texting, since older people often don't know all of these. Of course, this is a kind of "language" that older people can and probably should learn. My own sons have taught me very well!
As a general rule, whether the communication is writing, speaking, texting, or emailing, one aspect you will want to take away from your exploration of this is that everyone should be making adjustments, depending on the audience. We call this "code-switching." When we have a more formal relationship with another person, we speak to that person differently than we do with a person with whom we have an informal relationship. I assume you do not speak with your teacher the same way you speak with your friends. When we speak with a person who is not of our generation, we usually make adjustments for that, too. Similarly, we adjust for a person of a different culture. All communication requires some code-switching, which means that learning more about this in texting will help you to become more aware of how important this is across the board.