You're confusing two different forms, which have different meanings and uses. I'll try to show the difference.
First look at this:
a two-week holiday
This is a singular countable noun phrase. Since it is singular and countable, it needs an article, like any other singular countable noun phrase. The hyphenated bit is a compound word (think of it as one word with two parts), which is modifying the following noun. The following noun is the head of the noun phrase. The head word holiday means something like an experience. It is likely that you travel to a place, spend two weeks there having a nice time and then go back home. The holiday is the whole experience and two weeks is how long it lasts.
I had a two-week holiday in Brazil last summer.
Now look at this:
two weeks' holiday
This phrase has different grammar and different meaning. The difference in meaning is that holiday does not mean an experience like before. Here it can be understood to mean time without working. That's very different, and has nothing necessarily to do with travel. For that reason, it is not conceived as a 'thing' like before, in which case it is not a singular countable noun, and therefore does not need an indefinite article. The two weeks' part is not hyphenated because it is not a compound.
A good way to understand this form is to think that the apostrophe is saying 'of' or 'worth of'.
I have two weeks' holiday to take next month.
= I have two weeks (worth) of holiday to take next month.
= Next month I have two weeks during which I don't have to work.
Apart from with the word holiday, we also commonly use this form with the word time.
I'll see you in two weeks' time.
= I'll see you in two weeks (worth of time).
Finally, regarding your example with trip, I hope it's clear now that a trip can only signify an experience (just like a holiday), and not time (like holiday), which means that it can only be used as a singular countable noun, and cannot be used in the latter form above.
