Well, there're three things you've mentioned here: competitions, airshows, and Aero GP/Red Bull, which aren't really air races but more like travelling airshow acts.
Competitions. Have a look at the British Aerobatic Association website and it'll explain all you need to know about it. It's entirely at your own expense and initiative, but there are instructors, such as Alan Cassidy, who will train you and then let you rent his aircraft for a competition. There are four classes, Beginner, Standard, Intermediate, Advanced, and Unlimited, but Beginner doesn't really count (hence 4 real classes, even though I mentioned 5). Each class has a catalogue of maneuvers that are allowed/required in each and the competitions will be based on sequences made up of those figures. In each competition event there is a known sequence (published the previous autumn and used at each event all year), plus at least one unknown sequence published at an event. It's good fun, and a good way to give focus and purpose to your flying. Highly recommended as it puts you in touch with real expertise, but the competition side of it can be a bit naff at Standard as the rules are really kind of lax.
Airshows. Airshows are highly regulated and controlled by the CAA. Every pilot flying in one has been issued a Display Authorisation by the CAA for permission to fly in front of crowds. DAs are issued at various levels of difficulty and altitude based on your experience. First time out expect approval no higher than a Standard cataloge DA and no lower than 500 feet above the display line. As for costs, there really isn't much in the way of sponsorship except in the Unlimited category and that's for serious pilots with years of experience and seriously hot machines. As a PPL you are allowed to charge a nominal fee for displays that is directly related to defraying your costs to train and perform a display, usually around £300 or so per event, but that will unlikely really cover much of your costs. Expect to pay in one way or another for the privilege of displaying.
Aero GP/RedBull are both more like travelling airshow acts a bit like Formula 1. You're invited to join and the flying is pretty strictly controlled. I think the "competition" element is genuine, but its a pre-selected field of pilots invited to participate and they're usually national or world aerobatic champions or highly experienced airshow pilots who've been around for years and are well known.
The way to start it all is as mentioned in earlier posts. Find an instructor as near to you as possible and get into the details with them. If you're serious aobut it, competition at Standard Level is the way to start and see where you want to go once you've got a year or two of that under your belt.
Whatever you do, have fun, be safe, and be an ambassador for aviation.
Cheers,
Pitts2112