As others have said, join British Cycling. You need to be a Silver or Gold member to be able to get a race licence (£42 or £72 if you don't get a discount somewhere). You then need to buy the licence on top (£37). If you join a club first, there is often a discount on the BC membership for new-to-BC club members. Once you've joined, paid and put a photo up they will send out the licence within the week. You will also receive the BC handbook, which will tell you the rules of racing all the different disciplines. Most riders don't read the book, but there are useful bits for knowing what's what (I have read it cover to cover, but I'm a sad-sack commissaire).
British Cycling have an article for people wanting to get into road racing here:
https://www.britishcycling.org.uk/road/article/20131213-Get-into-road-racing-0?c=ENYou don't actually need a licence to race a BC event, but day membership is £5 per event for Silver/Gold members, and £10 for non-members or bronze members for regional events. A licence is required for any national level event.
There are 2 main types of road race - closed circuit and road race. Closed circuit is basically crit-style racing, either on a closed road (short section, lots of laps), or on a specific racing circuit. These are usually quite flat, and a good way to get into racing. If you get dropped out the back, the laps are usually small enough that the group will catch you again and you can see how long you can stay in the pack again (as long as lapped riders aren't interfering with the race outcome). Road racing is self-explanatory, but is very rarely on closed roads in the UK. The main race is protected by the convoy cars and the marshals will stop traffic at turns for those riders in the group.
Have a look on the British Cycling website for upcoming events. You'll start at Cat 4 - if you enter events online the system only lets you enter events you are eligible for. If there are some in your area, Cat 4 only, or 3/4 events are good starter events. Also evening closed circuit races. These are usually E/1/2/3/4 (i.e. everybody), and are fixed time events, rather than fixed distance. These tend to have a smaller field, and can be quite varied in rider ability.
I don't know the Yorkshire scene, but York has a closed circuit. I think Middlesbrough also has one. If you are able to race in the week, I'd definitely recommend trying a weekday circuit race to give your legs a test. It will give you an idea of what sort of other riders are about in your area.
