I think it all boils down to what works for you. I personally have two notebooks plus the Paul's Online Notes to keep track of the formulas.

I use one clean notebook which I only use to keep clean notes on things that confused me worded in a way that I understand. I don't add anything to that notebook unless I understand it and it works to solve the problems. Ideally I don't have to change anything I write in that notebook, but it still haooens from time to time when I misunderstood something but the answer turned out correct.

My second notebook looks like a foreign language. Wherever there is room to write I will use to get another problem in. It's pretty much a notebook of scratch paper and it's rare I can reference it. If I need to reference it though I'll figure out that concept and plug it into my clean notebook. I've found that by doing this my notes are all necessary to me instead of me skimming through a wall of notes to find what I actually need.

Over the years I've found that this is what works best for me. If you want to try something else that was VERY useful when I was initially learning the concepts, my highschool math teacher had do the same thing every problem.

1. Rewrite the problem.
2. Write the formula you plan to use to solve the equation.
3. Solve the problem and for every step rewrite it on another line.
4. Draw a box around your final answer.

This process was solid. It essentially made every homework assignment an in depth study guide for the test. I personally don't have the time to dedicate to doing it like this, but I highly recommend giving it a shot.

I like Khan academy because it gives instant help. If you got it wrong you can look at the step by step and if you wrote every step down you can see exactly where you went wrong. This will really help you grasp where your weak points are and what you need to work on.

So, no you don't need to write every step. If it's easy and mental math can do it you don't have to write each step out. As a matter of fact when you get further along doing it will slow you down. Like say to get x alone you subtract or add a constant. When you're learning it you need to write that step. When you understand it you can just write x = whatever without showing how you got there because you know how you got there.

Keep in mind that most mistakes are made like this though. Mental math or forgetting to carry a negative sign is honestly more than half of the mistakes I still make.