There are many accents in America just as there are many accents in the UK. You should learn the English of any country you plan to either live in or study in. An alternative solution is to learn International English. There are parts of America that talk with a near British/English accent.

Glad to see our American friends standing up for their language, I am sure they are correct that you would be better off learning American English for a trip to Mexico or the US. You mentioned travelling to Italy, I think perhaps they might regard this as a mistake. There are some classic problem confusions between British and American English, including: AmEng: sidewalk, BrEng: pavement AmEng: pavement, BrEng: road, roadway (this one can get you killed!) AmEng: fanny, BrEng buttocks/behind; in BrEng fanny means 'female pudenda' (you might get a slap in the face for using this inappropriately). In AmEng a fag is a homosexual male; in BrEng fag means cigarette. In AmEng 'ass' means various parts of the lower body, including buttocks, in BrEng this means 'donkey'. In AmEng 'bum' means a poor person living on the street, in BrEng it means 'buttocks' (again!). AmEng: the check (in a restaurant) is the bill in BrEng. In BrEng a cheque is something you might use to pay the bill. In AmEng a theater is somewhere you go.to.see a movie, whereas in BrEng it means somewhere you can see live entertainment - Shakespeare etc. Most of the people you meet in Italy will either not speak English or, if they do, not be particularly confused: waiters will bring you the bill whatever you say, I'm afraid.

It depends on where you plan to live and work as to which version of English you learn. If you plan to be in Europe, then by all means learn British English but on this side of the globe, American English is more common. I know that is not your question but I do want to clear up the matter that there is not a "better English." That is a myth. Please don't fall into that trap. It can easily lead into a racist type of attitude. It is just like saying that Spanish from Spain is better the Mexican Spanish. I dare say that the majority of Spanish speakers in the world would be offended by that statement. Just enjoy the language and have fun learning.