What is the right varient: to go out for the day or to go out for a day?
In Module 4 in New Cutting Edge Pre-Int Student Book by Longman they teach to use a set expression - to go out for the day. So is "to go out for a day" wrong? Google has found lots of websites with both varients but more with "to go out for the day".
The difference is simply the definite (the) or indefinite (a) article. In the sentence, "He wanted to go out for the day" he wants to skive off on that specific day. If he had said, "He wanted to go out for a day" it means he is just dreaming of getting away someday, not today, as he's too busy today.
When you come to visit next Wednesday, I'd like to take you out for the day shopping. (On Wednesday).
When you come to visit next Wednesday, I'd like to take you out for a day [of] shopping. (Someday during your visit, any day, that would be fun, don't you think? Let's try to plan that! The only way it really would mean Wednesday specifically, is if your visit was limited to one day, in which case either a or the would mean the same thing.)