To take...
For me, this one always uses "a".
To take a vacation.
However, if you change it to a noun (gerund) it can lose the "a":
Taking vacations is my favorite past-time.
I speak American English, so, it may be that "to take vacation" doesn't sound wrong to British English speakers.
To go on...
Both with and without "a" sound fine, but possibly with a small difference.
I went on a vacation last week. Where did you go?
I went on vacation last week. Oh, that explains why you weren't in the office.
To me, "to go on a vacation" seems more exciting than without a. I would use the second version, if I do not intend to tell people where I went, but rather, just want to tell them I was gone (although they might still ask anyway).
The perspective is a little different.
I went on a vacation to Italy. (the place you went to)
I went on vacation from work. (the place you left)
But you could also say,
I went on vacation to the Bahamas.
To be on...
Same as To go on..., the article just changes the emphasis a bit.
Hello, this is XYZ Management Services. How can I help you?
Could you transfer me to Susan?
I'm sorry, she's on vacation till next week, do you need help with your account?
No.
If there is a definite article in the title, you say it as part of the title:
Today, I would like to discuss The Bridge on the River Kwai with you.
However, if there isn't, you don't need to add one.
Today, I would like to discuss Kind Hearts and Coronets with you.
The only time you would see an added article would be if the film title is used as an adjective phrase modifying a noun that take an article. For example, you would say:
I have the soundtrack stuck in my head.
and:
I have the soundtrack to Kind Hearts and Coronets stuck in my head.
Therefore, if you use the title as an adjective phrase, you would also say:
I have the Kind Hearts and Coronets soundtrack stuck in my head.
But this is because "soundtrack" needs the article, not because the title does.
You should also be aware that, if there is an article in the title, when you use it as an adjective you would normally not repeat the adjective. So, for example, it would be more common to hear:
I have the Bridge on the River Kwai soundtrack stuck in my head.
than:
*I have the The Bridge on the River Kwai soundtrack stuck in my head.
Best Answer
If you are giving a complete sentence, you should use the article. "The plane arrived in the USA." Etc.
If you are giving the names of countries in a list or chart or some such where you are not using complete sentences, then you simply put "USA", like you would put "Germany" or "France".