When you go for something or somebody, you physically travel to get that thing.
Ex: I'm really hungry, so let's postpone any further work. I'll go for the pizza we ordered. Does anyone want anything else?
Ex: John has volunteered to go for more hot dogs, so call him if you want him to bring something else. He's at the shop now!
When you go for something, you choose or decide on that thing. You can use the phrase Go for it! if you want to encourage someone to do their very best.
Ex: Bob decided that he had to go for the gold medal in the decathlon, because his father had done so before him.
Ex: I saw Bob before the competition, and noticed that he seemed nervous. I told him not to worry and to just go for it!
You can also say that you go for something to show that you like it very much, or that you are choosing that thing over other options.
Ex: I really go for convertibles, especially if they're red. The only problem is that some people think that you must be arrogant if you drive a car like that.
If you write something down, you are writing it on a piece of paper.
Ex: If you remember the name of that book, write it down for me. I am not very good at remembering names.
Ex: If you need something from the principal, write down a formal request and submit it to his secretary in the morning.
You are also writing down if you have to write more simply than usual, for a less advanced reader.
Ex: Although he has a PhD, Frank wrote down to his readers for his newspaper column, since he knows most of them don't have the same level of education.