There's a few things you will need before you start hill climbing.
First and probably most obvious is a car. Theres a whole range of cars available out there and not very many that wouldn't get excepted for hillclimbing. I would suggest coming along to an event and having a look around, speak to a few people about their cars, everyone is pretty friendly. The other thing to remember is that the cars are all put in classes according to the CC of the engine. So a 1.4 wouldn't compete against a 2.0litre for example. You can run anything from your daily driver with an MOT to a full blown single seater, but I wouldn't suggest that for a first time.
Race licence, now this can be applied for online at Motorsport UK theres a link at the bottom of the page. The licences are pretty resonable and theres no need for an assessment just add your photo and send it off.
Equipment. once you receive you Licence you will get a 'Blue Book' which state all the regs and rules for the current season. In here you will find what is required.
The basics are Approved race overalls and a Motorsprt uk approved Helmet & Gloves
These can be found fairly cheap secondhand.
Join the club, by joining the club you get access to events over the country. Get advice and help from club members who have years of experience on building up a car or just competing in their daily cars. Notifications on up and coming events and where to book them.