Like in many countries, the USA has many different dialects and accents including the General American accent. Several years ago, a group at Harvard did an American dialect survey that further elucidated to many people the fact that some Americans pronounce differently-spelled words the same, such as cot-caught, pin-pen, tin-ten, and, my favorite, Mary/Merry/Marry. There are more, but those are a few common ones. If a learner of English wanted to focus on an American accent that differentiates the most words (least number of homophones?), then which accent should the American English learner focus on?
Being a native speaker of US English, I would recommend either a New York City or Philadelphia accent as having more-than-average phonological diversity.
New York City and Philadelphia accents typically distinguish:
- Mary/marry/merry (merged nearly everywhere outside of Pennsylvania, New Jersey, New York, New Orleans, and Boston)
- pin/pen (merged in most of the southeast US)
- Don/Dawn (merged in New England, some parts of Appalachia, and most of the Southwest and West other than San Francisco)
- plan it/planet (merged in most areas outside the Boston-Washington Megalopolis)
I usually use the Philadelphia accent. Note that the presence or absence of a merger does not seriously impede the ability to communicate - I barely even recognized that I spoke differently from a Californian until I was over 30. I knew that there were differences, but I did not recognize the nature of those differences.
Learning an accent with fewer mergers does not mean that you have a more expressive vocabulary or are able to express ideas more concisely. People adapt and do whatever they have to do to communicate! For example, in areas where pin and pen are merged, it is common for people to speak about "ink pins" and "stick pins" - they do not simply say "pin" and let people stay confused! They also have the ability to omit the clarifying term if it is clear from context, so one could even argue that a merged accent is more flexible in that a speaker has more options to choose the level of ambiguity to use.
By contrast, for an accent with a large number of mergers, check out the Valley Girl accent. Valley Girl speak even merges sentence tone, making declarative statements like, totally sound like questions?
Note that the mergers can and are used as Shibboleths in the USA. In fact, they are one of the things that people will notice the most, either consciously or subconsciously, when trying to place a speaker. One time, I heard someone talk about plans to "get merried". I instantly knew he was not from anywhere close to the US East Coast, and I was right!
Also, check out the song Mari-Mac from the Newfoundland band Great Big Sea. It's easy to sing if you merge Mary/marry/merry, wickedly hard if you don't (it becomes a tongue twister). Find your favorite rapper in Harlem or the Bronx and challenge them with this - instant laughs.
I don't think you'll really have much choice in the accent you can focus on, unless you're already living in a particular area. Most lessons in US English will teach the most standard "General American" accent. It's possible that your instructor(s) may have a regional accent, but they'll generally strive to be as accent-neutral as possible.
If you're living someplace (e.g., Boston or Texas) that has a stronger regional accent, you'll probably pick up some aspects of that accent, but I don't think there are any specific courses in Texas English for Speakers of Other Languages.
If you're a non-native speaker, you'll probably have your own accent already, which can be very difficult to lose entirely.