They're not nearly as different as you think. Less than fifty words are different between American and British English and there are also a few other minor spelling differences (these were only made because the Americans wanted to seem less British). They have the exact same grammar.
The only instance that I can think of where a spelling difference reflects a pronunciation difference is in the word aluminium/aluminum.
The only difference between standard American and British pronunciation is a slight accent difference - they say their vowels slightly differently (but they'll still sound like the same sound) and different emphasis is put on 'r' (Americans pronounce it heavily (think of pirates) and at the ends of words, whereas British people say it less heavily and omit it at the ends of words). If you ha e trouble saying the English 'r', the standard British accent will definitely be easier for you.
There are of course regional accents within the US and UK. Standard British and American accents will sound almost identical to you in comparison to, for example, a Texan or a Cockney accent. Most actors in American movies adopt a mixed American-British accent so that all Americans can easily understand them (the idea is that standard British accents are considered to sound clearer and crisper, even to Americans!).
You should also realise that when you speak English to a native speaker, they probably won't know what kind of English you were taught until you tell them (because you'll have a French accent instead of an American or English one).