Hi Joel,
Like many others, I taught myself*, with a lot of help from this DeeperBlue forum & its members and a couple of short but useful chats with an experienced SA spearo, who used to live near you (Sunbury-on-Thames) - but I am pretty sure he moved out of London since then (to be nearer the coast). Bad news is that much of the kit is personal, you will probably have to buy it yourself and you will need most of it immediately. The good news is that it need not be that expensive - that's up to you - and it lasts a long time, as long as you don't loose it and look after it.
Regarding clubs, there are 2 or 3 near you - lucky you! The most convenient is probably the
London International Club:
https://en-gb.facebook.com/LICSpearfishing I believe they organise trips, competitions and events - so a good bet for you. Your best bet might be to find an experienced spearo dive buddy.
I believe there is at least one Sussex club, pos. 2. Try looking up the
British Spearing Association for info. on clubs:
http://www.underwaterfishing.co.uk/
Individuals and spearfishing equipment resellers sometimes offer training courses. Importers of RA spearing gear have offered training at different times in the past (in Portland, Dorset or Cornwall) - you might try googling spearfishing training. Such courses may be able to provide some but probably not all gear; however they will often provide valuable guidance and discounts - not buying unsuitable gear can save quite a lot of money.
*This "teach yourself" approach will suit some people but not others. It is not without risk. I should mention that I had done quite a lot of open water swimming before that.