Well, you can take inspiration from pictures of texts that are posted all the time.
Like this:
Sorry for the language, it's just the first one I found. But basically, online, people are accustomed to reading text messages in this format. Doing this in book form would mean indentation, and so on, if you wanted a full convo.
If not, you can include it in the body of the text, in short by paraphrasing.
Book editing hasn't exactly caught up with this, so the "right" way looks just...wrong.
Here's a handy link to help with that.
Italics, beats thrown in, and tags here and there seems to be the best way to do it, unless you are doing a convo in its entirety. As long as you are consistent in your format, it will be fine. On the internet when relaying texts not through pictures, the format is often:
Me: I don't know when I will be back
Mom: well dont stay out all nite
The punctuation problem, or lack thereof in REAL texts is just something you will have to decide. If you want to be authentic, you can keep them as they are (shown above) or you can correct them. This happens in dialogue as well. In real life there are a lot of ums, ands, likes, repetitions and speakers interrupting themselves. In book conversations, we throw a few in for realism, but take out most for the sake of readability and art.
This is such a new area that there isn't really a consensus. Some publishing companies have their own in-house standards, which is really only something you'll come up against if your book is text-heavy and/or they actually pick up your book. But mostly, writers have to set their own standards and stick with them within the bounds of their novel.