You can just say the MET to people who know it's a museum (or a test or an exam), but to people who aren't familiar with these things you might want to add the last word so they are clear what you're talking about. Hope this helps :)

People say it both ways.

Hi, you can just say the name (without company/museum etc.). Some examples: I'm going to the MET. I work at Google, previously I worked at Apple. I'm doing my HSC. I passed my IB. I've completed my MBA.

As Kai_Aoel said, it depends who you are talking to and whether you think they are familiar with the topic. With my English students, I can just refer to 'the IELTS' and they understand that I mean the IELTS test. However, if I'm talking to a friend, I would say 'the IELTS test'. The same goes with the HSK. When I talk about it with other people studying Chinese, I can say 'the HSK' but with my family I would have to offer more information and say 'the HSK test'.

‘The Met’ isn’t an abbreviation and shouldn’t be capitalized. It’s just a colloquial shortening of the full name ‘The Metropolitan Museum of Art of New York’ If somebody doesn’t know what it is, you’d say ‘The Metropolitan Museum of Art’ the first time, and then ‘The Met’ once they do. You should definitely visit the Metropolitan Museum of Art - we call it The Met- while you’re here.