Joining a club isn't essential, but is a good idea. For races that are oversubscribed then local club riders will normally get preference. I started off racing last year without being in a club, but only managed to get into 1 of the first 3 British Cycling races I entered. I joined a club and haven't had an entry turned down since.
If you do join a club you don't have to ride with them every week. I've only been out on one Sunday club run as I'd rather go out by myself and do 50 miles in 3 hours and be back for lunch than take the best part of a day with the club. I have however been out on quite a few weekday training rides - these are faster rides and only those training properly to race or time trial do them.
To do British Cycling sanctioned races you will need at least a Silver level BC licence and a race licence. You could also look at doing some
TLI events - these are cheaper to enter (you need a TLI licence though, but its only costs a few quid) and a good introduction to racing as they're a little easier than BC events IME. Often they are run as handicap events with 2 or 3 groups - the slower riders go off first with a few minutes advantage. The result is that until the groups come together there's less attacking as you need to work together to stay away or catch the other group.
One word of warning though - don't underestimate how competitive even the lowest level (3rd/4th Category) racing is. Average speeds are usually up around 24-25mph and you need to get used to making yourself really suffer for a few minutes as the pace is often all over the place.
