Which English do you want to speak? There isn't one, but several Englishes that people all over the world use to communicate. American English certainly dominates, being the language of Hollywood, computing and aviation. Some would even go as far as calling their language "American", and why not? It has several important differences with my English, British. But for a foreign learner to say that they speak English and American is exaggerating somewhat. The biggest differences between British and American English are vocabulary items, just like there are regional differences in any language.
A French speaker is likely to get confused when there is the added problem of "false friends".
A good example is the British word "chips". Here in the UK, chips are fried potatoes, generally cut a little thicker than their American equivalents, "French Fries", which are not French at all, but Belgian. However, in France we use the word "chips" for thin slices of fried potato that come in a packet and are eaten as apetisers or aperitif.
A French teenager may be very proud of his new "baskets"! I would use a basket to bring home my vegetables from the market (panier). The French word has come from the sport basketball, while the British say "trainers" (shoes for training) while the Americans say "sneakers" (not to be confused with "Snickers", the chocolate bar). "To sneak" means to walk about silently, as if you were somewhere you shouldn't be.
As for grammar, the good news is that Americans use less and less the dreaded present perfect - so you have one less thing to worry about. Words like "just" "ever" and "already" can be used with the past simple, whereas in English it's the present perfect.
British: have you already seen this film?
American: did you already see this film?
The British generally have no problem with American English as they are used to watching American films (or movies if you prefer). The Yanks, on the other hand may have a few difficulties understanding a Briton, especially if he uses slang words.