Hi Joanna
Start with your face, no doubt about it. If you have dark hair you can start with laser and see how you respond. Some people are luckier than others. If you have light hair then it's the joy of electrolysis, about 18-24 months of weekly torture.
HRT often knocks out body hair - torso and limbs - but not with great speed. You may need laser on those areas as well.
Hair removal is probably going to equal or exceed the cost for my SRS, and I'm one the minimally haired people. I suggest you find a good place, close to home, and negotiate a long-term discount once you're happy with how they work. For example, I have 1,200 minutes of electro for €2,400, which is a good price where I live.
You could consider one of the very few places that does a whole-day electro session on your face. I know a few girls who've done this. It's an absolute torture, but would likely clear your face in around 8 sessions. The downside is that you have a beard by the time you're ready for a session. Urgh!
So yes, in answering your question, start electrolysis yesterday, or last year even

Hugs
Julia