How is it known as?
I am interested in learning to talk exactly like all of the following:
British, Russian, Greek, Cuban, Spain, Argentine, Colombian, Puerto Rico, Dominican Republic, etc.
Speciality within linguistics.
There is no one way that people from any of those countries talk.
Wait, you want to learn to speak English with the accents associated with people from those countries? That’s called dialect training and it’s part of some theater programs. There’s no academic subject in which you learn to mimic foreign accents. (Weird Q, if I interpreted it correctly)
You have multiple posts indicating a broad range of interests – acting, baking, journalism, and forensics. You’re ~23, right? Have you attended college at all yet? Maybe you should pay a visit to your local community college and see what type of programs they offer. A counselor can help you decide how to get started. You can take the required general education courses while you figure out what major you want.
A phoneticist. Like Henry Higgins from My Fair Lady!
Well, linguistics doesn’t teach you all the specific features of specific dialects, but you do work with data sets and start noticing those details when you hear people talk. There is a subfield called dialectology that directly studies dialects though. But for that you’re studying the dialects themselves, not how to reproduce them. Even if they did focus on reproducing a dialect, you wouldn’t be specializing in that during undergrad. And even if you could, you’d be focusing on very specific dialects…to be perfectly honest, you’d be focusing on one specific dialect if you wanted to go into a decent amount of depth on it.
That said, if this is a question pertaining to majors, don’t even think about majoring in linguistics if just learning how to imitate accents is your intention. For that you would want something pertaining to dialect training, which marvin100 touched on.
I don’t want to major on it but kind of use it for Acting. This is what I want to be capable of doing: https://www.youtube.com/watch?v=w43__wzPItg
Then as @marvin100 said you want a dialect coach. Actors use them.
Also note that most actors don’t have a range of 9+ different countries’ dialects. Some actors can do one or two countries’ or regions’ accents pretty well but not really anything else. Some actors are terrible at dialect but get cast in movies anyway because they are big box office draws (e.g., Leonardo DiCaprio’s terrible Afrikaans/South African accent in Blood Diamonds.)
Other times casting agents don’t care about fine distinctions - they will hire a Colombian to play a Puerto Rican any day and vice versa (see Zoe Saldana, who is Dominican, Puerto Rican, Haitian and Lebanese but was hired to play a woman of Colombian descent in Colombiana, and speaks with an American accent. She also played a Bolivian woman in The Losers).