You might find you don't like english, so I wouldn't go ahead investing in breeches, expensive shoes etc. Most riding schools have helmets and shoes if you don't have your own - they're probably not very nice bit they'll do the job.
Also, wouldn't go straight for breeches and half chaps. Get a pair of simple jodhpur boots with elastic on the side to start with. All they need is a flat sole and a heel, you can pick them up pretty cheap in most tack stores. You can just ride in tight pants - that's what I do anyway and I've ridden english for years. I have nice jodhpurs that I wear to shows or training things but just for mucking around I just wear $10 tights. No real seam, black so they don't get dirty and they aren't very hot. They also have heaps of give. They might wear out quicker but in the long run they work out cheaper, because jodhpurs never seem to last me that long, and they always look a bit funny.
Its weird but once you ride with half-chaps/gaiters all the time its hard to ride without them. They sort of hold your leg in place without you needing to learn how to, and I notice people who have learnt in them don't seem to have such a firm leg position, learn without them and then if you want/need them invest in some good ones otherwise you'll go through them very quickly. I have a pair of Ariat boots and gaiter/chap things and I love them and they have lasted me a fair while, but I'd go through a few cheap ones a year.
You'll probably want gloves pretty soon, in english you ride with both hands and generally work on a light contact, but school horses have this terrible habit of ripping reins out of peoples hands and it hurts. They are worth getting a good, stretchy pair too. The ones with the little lumps for $5-$10 are terrible.
Besides that, wear a loose-ish t-shirt/polo shirt/jumer (not flappy loose), tie your hair back and that's about it.
Once you start riding you'll find out what you need.