It's interesting, as I have been asked about what qualifications I have, and so have local teaching friends. I agree that word of mouth recommendation is extremely powerful (as with most things) but I do also think that there seems to be a degree of local variation as well in terms of what works best and what doesn't. I agree with BadStrad about getting your name known locally and any playing opportunities are a brilliant idea. (Although I would have to add to that the best performers don't necessarily always make the best teachers!)
Local music shop- I wish- we haven't had one of those for years here! It did use to be a helpful source of new business. But over the years I've found that having a website, and making sure I'm on the free listings sites, netmums, and the local library database have been helpful. I don't use Facebook for my teaching at all at present but I do know other teachers have found it helpful.
Studying for a diploma is a longer term commitment anyway, so not something which would directly bring in new business straight away, I'd have thought- I'd say go for it if it's something you want to do for yourself, and for longer term benefits to your business, but research other options as people have suggested.