There are more American accents than British accents. This is largely because America is simply better in every respect than so-called "Great" Britain, but also because Americans are inherently more creative. Pop quiz: If Britain is so great, why don't they have any more colonies?
Anyway, if you'd like to learn more about British accents I suggest looking up "English Language" in wikipedia and there should be a section in there about different dialects of English. There are articles for each one, and you can even learn exactly how the pronunciation differs, as well as special words and grammar that apply to each one, so that you can know the local terms wherever you go!
Here is the dialect list, there are links to many different parts of Britain backing up what Cane said about many accents. It's not British, but I find the Singaporean English fascinating. It has a huge amount of Indian and Southern Chinese influence, not just on vocab but sentence construction as well, at some points the patois is only barely recognizable as English! But just remember that an American accent is even better, it will make you sound like a real cowboy, like John Wayne or John Cusack!
