Which dialect you choose to study should be based on what you want to do with English and where you want to do it. If you live in a country that has a historical connection to Britain, you should learn British English. The BBC World Service and bbc.co.uk are useful resources, as are BBC programs like Doctor Who and QI.
If you want to learn English in order to work or study in the US or to work in your country for an American company (for a few examples of language goals), I recommend learning American English. To put another commenter's answer differently, Americans are much more culturally insular than the British are. Most are entirely content to never leave the US, and interaction between you would be difficult without American English.
Standard English English by definition is NOT a dialect.
If you're in Europe and dealing with British people you are well advised to not blindly copy American habits. It's not simply accent but politeness being expected.
OTOH having spent time in America, a little work is required to avoid confusing them, Americans are much less aware of other cultures in my opinion. I'd meet many who had never left the mainland USA, ever.
From what I've seen good courses expose you to both, just viewing shows and looking around the web, you'll assimilate American English.
I should focus on speaking a Standard English approved by top level universities, it's the local dialects which vary more really.