The Open University does a Diploma in Literature and Creative Writing that you could study part-time at home and which could also constitute part of a degree in, say, Literature or English Language studies. They also do an introductory short course in creative writing that you could try out as a taster.
Diploma here:
http://www3.open.ac.uk/courses/bin/p12.dll?Q01 E25
Short course here:
http://www3.open.ac.uk/courses/bin/p12.dll?C01 A174
I've done the short course and I can tell you it's a lot of fun and very useful.
If your aspirations veer towards a full time course, then a number of universities now offer degree courses in creative writing and related subjects.
You
could enter a few competitions, but they're not likely to make you famous. For that matter, neither will a course of study, but the latter will point you in the right direction and bring you into contact with published authors (Andrew Motion teaches the Diploma course at Leicester Uni). I'm still not convinced of the usefulness of such courses at university level. Different people say different things about them, as you'd imagine, but really you need to make up your own mind on this.
The other way is to join a writers' group - actual or virtual - and let others see your work. Again, not guaranteed, but you will get feedback on what you've done.
http://www.writersdock.org is one of the best sites around - lots of pros and guest contributors around for advice, as well ordinary mortals also trying for the top.
From there, it's a matter of submitting your work to agents and publishers and getting rejected dozens of times before someone s