First, I suggest you try shooting yourself before diving in and trying to get certificates (permits) and stuff.
On October 14th there is an open day at the National Shooting Center in Bisley. Here you will be able to try lots of different shooting disciplines including clay shooting, gallery rifle, black powder pistol, sporting rifle, target rifle, practical rifle and classic/historic rifle. Many gun clubs around the conutry offer open days, sometimes regularily e.g. once a month, othertimes less so, perhaps twice a year.
Unless you are at an open day or other special arrangement, you can't just turn up and shoot without a certificate. You need to apply at the club for a probationary membership. They will give you a form to fill out and you'll need at least one referee to sign it also. I can't remember if you need photos or not... If you do go shooting and like it and want to carry on, you'll first need to become a probationary member of a club as above.
Assuming this is granted, you'll have to attent the club regularily for the next 3 to 6 months. Different clubs work in different ways. Some of them have a specific day each week for probationary members to attend, others will let you turn up whenever. On probationary membership you're assesed for various things and also taught how to shoot safely and how to use the range and follow procedures etc.
Assuming your probationary membership is succesful, you will then be accepted as a full member of the club.
Once you are a full member, you can apply to the police for a firearms certificate. This costs £50 and can take anywhere from a couple of weeks to a year to come through. A firearms certificate is what allows you to own firearms. You'll get a form either printed off the web or from a police station. Fill it out, you'll need to give a 'good reason' for your application. This should be 'target shooting'. You'll also need two referees and four identical photographs. Make sure you complete it perfectly and phone your local firearms officer if you have any problems. Any mistakes will likely double the time it takes to get granted as it will have ot be sent back to you then back to them etc. Once you've sent it off, phone up every other week and ask how it's going - after a while they'll say 'sod it' and do yours as quicky as possible. If you make no mistakes, it should come through reasoanbly quickly, perhaps a month or three.
During the process of applying for a firearms certificate (FAC) a police officer will visit your home (at a time arranged between you that is convenient for both parties) and assess the security of the home. If it is not secure enough, you will be given advice on how to make it so. If this advice is not followed, you cannot be granted an FAC. You'll also need to explain how you will sotre the firearm - a gun room or gun cabinet in a secure place like a bedroom is usually what you'll do.
At first, only apply for one firearm and don't dive in for something like a long barreled revolver which is only borderline legal. Make it something sedate, suitable for a newbie, like a Ruger 10/22 or a .303 target rifle.
Common types of weapon allowed on an FAC are semi auto and pump action shotguns, lever action rifles, bolt action rifles, black powder rifles, black powder pistols, straight-pull rifles, single-shot rifles, long barreled revolvers and semi auto .22 RF rifles. There are also some types of pistol allowed for collectors.
Some types of shotgun (break-open types and also semis or pumps with a magazine that holds no more than two rounds) aswell as smooth-bore muskets do not fall under an FAC. Instead, they fall under a shotgun certificate (SGC) which is similar to an FAC but a bit easier and quicker to get hold of and not quite so restrictive.