I remember this stage well, two sons both growing like weeds, between 12 and 16 they each grew over 40cm in height, and both constantly needing outdoor stuff
My local outdoor shops, Tiso and Black's, stock kids and young adults size clothing and kit. Mind you it's not cheap and at your age you'll grow out of stuff so fast it's really not worth spending a fortune on something. These shops have branches all over now.
A backpack as you'd use for school will do fine for a short outdoors trip; Berghaus make good small sized/adjustable ones as do North Face, Lowe Alpine and Jack Wolfskin if you're after something a bit more substantial.
Buy the pair of boots that fits you best (army cadets/ air cadets boots are stocked by Arco who sell industial clothing and have branches nationwide) and never mind the name.
Waterproofs come in so many different qualities that it's worth your while thinking about how much use you'll give them. If the answer is "lots", then the folk who sell clothing for sailing are well worth a look, it's not all dayglo coloured
They have an excellent range of clothing for children and small adults.
Fishing outfitters are maybe a better source and they are usually a good bit cheaper. If your budget can stand it I found the hunting & shooting clothiers had some good stuff for the boys.
One of my friends has just bought a set of green waterproofs for her 14 year old, in Millets, that claim to be breathable, for well under £20.
Army surplus type stores are worth a look too, especially if your budget is tight, and there's always ebay.
Best of luck with it.
This is a good thread, there're a lot of parents on the list who'd be interested in good kids sized kit.
Cheers,
Toddy