I learned myself to not mouth breath, I had to as I couldn't get a Full-Faced mask to work for me.
I did it without a chin strap as well, the only problem I found was that air leaked back out of my mouth, even though it was shut, it would get out past my lips. A chin strap did help a bit, but early morning about two hours before I was due to get up the small amount or air that did get into my mouth but not out would dry it out just enough so that it would get round inside my cheeks (of my mouth) and manage to get out past my lips. Once that happened it was not a lot that got out, but enough to get a very dry mouth and this only made things worse.
I never found a cure for this, but if I took a drink of water, it would stop. At the higher pressures I was at with my old machine I had to turn it off, take a drink (just enough to wet my mouth) and then start the machine again.
However, with this new machine, I can use one of those small bottles with the pop up top and keep the machine running and go straight back to sleep.
The simple answer is that you can teach yourself to breath through your nose, though you have to try to do it all the time.
If you have any problem with your nose, you might find it difficult to do, if not impossible. I used to mouth breath nearly all the time, never thought anything about it until I went to the dentist one day and he asked "Do you always breathe through your mouth?" Never thought about it, but this was just after I got a Nasal Mask from the Sleep Clinic when I was first diagnosed. I couldn't wait to ask why they gave me a Nasal Mask when I was a moth breather.
I can't remember what their response was, but the first mask I got was almost all seal, it used to float around on to of your face and was hopeless, every time I moved it would break the seal and air would fly everywhere!
When I think of some of the crap masks that were around here in the early day I am surprised we got any sleep for fighting with the masks.