So, it's entirely possible to learn "English" in a general sense, especially if you want to use it for more formal things like studying an academic subject or doing business or international aid work. The differences between the dialects aren't as big a deal in those contexts. But if you want to visit either country, and/or you are interested in either *culture*, you should try to learn the specifics of each dialect on its own. There are straightforward vocabulary differences and then there are differences in how likely speakers of each dialect are to use different terms. There are differences in phrasing and grammar to the point that certain things aren't said in one dialect or the other. There are differences in social use of language, with different attitudes towards language. Brits and Americans, in general, have different writing styles that reflect how kids in each country are or aren't taught to be literate. You *could* learn that the Americans say "elevator" and the British say "lift" at the same time, and keep that straight. Just a matter of how good your memory is. But it might be harder to get a sense that the Americans love commas and coining compound nouns, while the British tend towards different vocabulary choices, more use of hyphens, and perhaps more meandering sentences. And whether that stuff matters to you depends on how well you want to know English. So it's a question of why and how you want to learn English and what you'll use it to do.

Those aren't 2 different languages, it's the same language with slight vocabulary and prononciation variations. Just learn English.

Less than 30% of the world's English speakers speak it natively. Of those, 70% are in the US. Vocabulary and grammar are similar. But keep in mind that speech is another story. For example, an American will understand many Swedes (Norwegians, Dutch etc) speaking English more easily than many British speaking English.

Very few words are different, it's more a question of accent than vocabulary.

*some idioms* I'm not a native speaker, even though one of the biggest differences is that American English uses more the /r/ phoneme, whereas British uses more the /t/ phoneme. Just look at words like mother, what, water, or, and so on. In my opinion, the only way to understand those differences of pronunciation, is to learn phonemes.