In reply to samben6:
Hey, yeah I agree with what others have said, clubs are great and any of the courses are excellent I'm sure.
I'd check things like Scouts (it would be Explorers for age 16) and cadets too. I was part of an Explorer unit in London and I learned an awful lot, and it put me in touch with a lot of similar people who also enjoy getting outdoors. Explorer units in general are a very mixed bag, and the vast majority don't do particularly adventurous things, so there might not be one locally that offers what you're looking for! A friend was part of Cadets around 10 years ago and he had access to a bunch of subsidised courses, and got a week's winter hillwalking course for next to nothing. I have no idea how much that's changed over the last 10 years tho...
A few companies also offer courses for u18s (Pure Outdoor in Peak District is one, and Glenmore lodge does some specific u18 stuff too), and these can be great, and if you grab a mate who's over 18, you can hire out a guide for the day (most understandably want an over 18 in the group in my experience). If you get a few mates to split the cost between, it can be more affordable than it initially seems.
I learned a lot through youtube and books too. It's not a replacement for proper instruction but can make limited hours of instruction stretch further. If you go on the Mountain Training website, they have a bunch of books which are excellent and comprehensive guides to skills for hillwalking, climbing and mountaineering. Freedom of the hills is another classic, although I found it almost TOO overwhelming content-wise.
For the mountaineering side of things, hillwalking can be a great stepping stone towards that, and it's just more accessible. It teaches you a load of really important skills for the more serious stuff, stuff like navigation, especially in poor weather, and keeping warm in bad conditions etc.
Finding people to go with is tough. If you're short on outdoorsy mates at school/college, try chatting to people at your climbing wall? Even if you don't end up going on trips with them, I've met some very experienced people who are always happy to give out advice on things, which defo has helped me.
Transport wise, it doesn't need to cost a bomb. You can grab a 16-17 saver railcard which can get you half-price train tickets, then you can split tickets to save even more money (at the cost of a slower journey) and you can get trains to Wales, Peaks, Lakes etc for really good prices (I paid £11 from London to Penrith just outside the lakes).
I'm 18 and I remember feeling super frustrated when I was 16 for the same reasons. It feels like most things (like mountaineering clubs, JCMT etc) only cater for over 18s, and a lot of the "youth" clubs stop at 16, so it's not as easy as it should be to get into the outdoors as it should be!