It depends upon the game of course. For some games I will hardly take any notes. For others I will practically write a book. Right now I am actually halfway through the process of re-writing a walk-through for the original Wild Arms since it can be difficult at certain points. I originally wrote it for my own use, but when I'm done with the re-write I'm going to give it to a friend who wants to play the game.
I usually don't write things down. Things get stored in the vault and I don't even realize it until it's time to access them Even when replaying a game. I'm currently working through Wild ARMs Alter Code: F and I'm noticing some of the small differences in the stories/dungeons as well as some of the similarities.
Sometimes I'll write down a quote I think is funny or if some super big hint involving numbers and/or letters just came up and I'm too tired to keep playing or have to go somewhere.
I've also been known to work things out on a piece of paper. The first example I can think of is from Parasite Eve 2. There was a puzzle involving a computer virus and alphabetical order. I saved myself a headache by writing it down I also used to write thing down for one of the casino games in DQVII. If I recall correctly, it was some form of the game "memory" where you try to match two cards that are the same. I won several times and got a whole host of extra items.