It's absolutely crucial to record blocking. The beauty of live theatre is in the sheer number of variables that can and do change from show to show and from actor to actor - no two performances will be identical. Movies can't do that. TV tries, but cannot capture the urgency of it all going down in the same room with the viewer. However, variables are pesky and unreliable critters. The best defense in keeping a beautiful, well-coordinated symphony of performances from descending into utter chaos is the stage manager with their book.
Things do go wrong. Scripts get lost, people miss rehearsals, and blocking evolves from rehearsal to rehearsal. Even the most well-organized director is not going to remember every single bit of motion that he's assigned to every single actor in the cast. The best actors in the world still get a little wobbly with their placement onstage once you take the script out of their hands. Even if you have a photographic memory for blocking, you are still a variable. If you and your lead actor were in a horrible accident on the way to rehearsal, perhaps in high school they would postpone the performance. In professional venues? Fat chance. They'd find a new stage manager, fly up the understudy and keep right on trucking - if your book is in good enough shape for the new folks to figure out what's going on.
This doesn't even tap into what happens six months down the road when your school decides to remount the show for the all-state festival, or another company wants to restage your version of the show - your book will become the only record of how it was done the first time. Or what happens when your director doesn't attend performances for weeks on end and it falls to you to make sure the show looks the same - exactly the same - every single time.
So yes, blocking in the prompt book is a must.
As for how? For the stage manager, convenience and tradition always take a backseat to what works best for your show. While we get finicky on this website about tidy paperwork and trading pointers in prompt book artistry, that's all frosting. The real cake is keeping the information and details on hand so that the people onstage can keep being productive. While having a sharp, svelte prompt book will help keep you in a better mental state, all that really matters is if a) you can read it, b) you can use it efficiently and in a helpful manner, and c) if someone else can read it effectively in the event that you are taken out of commission. Get the function down first before you worry about the form.
There's definitely some conventional shorthand and diagramming that stays similar across all prompt scripts, but as the infinite dialogues about prompt book setup seen across SMNetwork show, no two prompt books will be identical. If you don't have enough room to write blocking, make more room. As I'm right-handed, I usually started with the script printed on only one side of the page - usually the left hand side of the binder- with blank space on the right for blocking. If that isn't enough room, add more pages to the flap on the right and fold them over so that your book still closes flush. Or, staple extra pages to the top of the right hand side to make a mini notebook for each page of the script. Make as much room as you need. The one thing I don't recommend is spacing out the lines of the actual script - you'll want to have your pagination be identical to whatever the actors have, so that if someone says "turn to page X," you wind up in the same place as everyone else.
I've seen prompt scripts for particularly elaborate staging that wind up looking like an origami nightmare of extra, added pages. I've seen others with handwriting so tiny as to require magnifying glasses. I've seen (and used) the two-script setup that Adam describes above - usually for musicals and/or large casts. I've even seen condensed, blocking-only scripts for running understudy rehearsals with only the bare minimum of the actual script included, but every single bit of movement. Given that you're still in school, I think it behooves you to experiment a bit while your mind is still open, before you get bogged down with "this is how your prompt book must be, Amen." Definitely get the blocking down. How you do it is up to you.