I'm sure you can find better info if you look up the formal sources, but in the mean time, from the perspective of someone who just saw TrgOs as fellow officers:
Firstly I think you have to have a degree in Education/PGCE of some sort, never met a TrgO who didn't.
There are few roles overseas, indeed I was aware of some criticism of the branch about 2 years ago because they had no obvious deployment role, and the general deal in the RAF was if there wasn't a deployed role, they shouldn't be uniformed posts, so Trg Os were looking at ways of being more useful in wars!! However, they are not all at Training bases in the UK, all Stations will have a TrgO, the larger ones may have 2 or 3. In general the Trg O is the expert, so does things like course design, approval, standardisation etc. The RAF tends to make the people who do, teach, ie aircrew teach aircrew, engineers teach engineers, the TrgO makes sure the courses are comprehensive, have the right resources and all sorts of technical Training stuff like doing training needs analysis of jobs, looking at training gaps etc.
They also have a key role in getting airmen the qualifications they need for promotion and commissioning, and everyone the qualifications that they want, eg ECDL, GCSEs, A levels or whatever.
They also usually oversee the Stn Library and careers centre on a Stn. Probably head up the Commissioning Club - and possibly many other things I never noticed them doing. Always very useful when I went to see them on Stn. I saw a few working at HQ level, can't think of a duller job, but then I'm not trained to be a training officer, so I wouldn't know, would I!
Basically if you have a passion for offering educational and training opportunities to adults in a military context, it's the job for you, otherwise Provost is a million times more interesting!