First, let me thank everyone who kindly answered my questions in the forums here about how to do this efficiently. Now that I've mastered the skill, I felt I should summarize it in this guide.
Sharing the information you get in a text message is easy, but not always ovvious. And to do it rapidly takes practice.
Let's say your spouse is driving and needs a friend's address. You text the friend and they send you an address, the combination for their front door, a link to a google map, a phone number and some information about finding the location. They also send a picture of the place!
This is a lot of stuff to retype in to another text for your spouse.
So you forward the text. Do it by double-tap and hold, and from the menu that appears, select more, and from the next menu, select forward, enter your spouse's mobile number and the message will be forwarded. The attached photo will forward as well.
Doubletap and hold is simple. A sighted person would simply single tap and leave his finger held down on the screen until that menu pops up. As a VoiceOver user, you perform the same double-tap you always do to activate something, but instead of lifting the finger on that second tap, leave it down until you hear the menu pop-up sound-- usually about a half second later.
What appears is a lot like the usual "share" menu; forward visually is an arrow, and voiceover speaks the word "forward" which is near the end of the sharing choices.
The one problem you'll have is when you double-tap and hold, sometimes you get a sharing menu for items in the actual message. If there is a phone number, it will suggest you call or text it. If there is an email address, you might be prompted to send email and if there is a link, it might open in Safari or you'll get an "open in" menyu for links or GPS coordinates. VoiceOver thinks you are focused on the link or the phone number or the gps coordinates so it wants to launch the phone app, or the maps app or the email app and you are trying to forward the entire message, not just act on portions of its content.
So in that case, your best choice is to rotor over to actions (you should hear "actions available") instead of doing the double-tap and hold. If you are focused on the actual text of the message rather than a link or the attached photo, then you'll get a sharing menu for the entire message. If this is tricky, rotor first to word navigation, flick up and down until you find a good word, rotor over to actions, flick up or down to select "more, and then left and right to select Forward.
Now, let's imagine another scenario. You have that same text from your friend, but you want to email it to someone without the attached picture and the combination for their front door! Instead of forwarding the text, a better bet is to select all the text, compose a new email and paste it in. You can do this the long way by rotoring to "edit" flicking down to "select all" and double-tapping, then flicking down to "copy" and doubletapping. Then you open a new email, and in its text body, you rotor over to edit, flick down to paste and that will insert just the text. You have to manually delete the door combination but that's easier than copying all the information from the text.
But there's a cool shortcut. VoiceOver can copy the last chunk of spoken information to the clipboard. So navigate to the message, and if you want the entire text, it will be read. Or navigate through the message, say by words, if you only want one of them. Then do a three-finger quadruple tap. That is tap with three fingers four times. VoiceOver will repeat the lasgt chunk of spoken text, and then say "copied to pasteboard".
If you have trouble with this three-finger quadruple tap gesture, consider slowing down VoiceOver's double-tap time-out. I had mine set very low at 0.2 of a second; that's just a tenth. In my quiet office or house, that worked fine. But in a moving, bouncing vehicle, which is where I often get texts I need to share parts of, I found I needed to lengthen it to 0.5 seconds, or nearly half a second. I did this by going to settings, general, accessibility, VoiceOver; near the end of all the voiceover settings is the entry for "double-tap time-out" which will open another screen with the fraction of a second currently set and below it increment and decrement buttons.
Remember too you can also rotor to "select" and flick up or down for word selection or line selection and left and right flick to select more or less. This is best practiced in the notes app, where you can tease out just the portion of text you want and practice pasting it in to a new note.
You should practice these features by sharing portions of email or texts you get with yourself; simply send yourself your very own self-generated spam. Imagine a scenario where you need to text an important gps coordinate, street name or passcode to someone in a hurry, and practice until you can do it smoothly and fast. One day you'll appreciate the time you spent mastering this skill -- I sure did when I had about thirty seconds to text a taxi driver before he stranded me.