What format should I write my cut list in? Should I add up the total amount of wood, or should I try to pick it up piece by piece?
You'll probably want to do both.
You certainly want to separate out your plywood from your solid wood. With your plywood, you can lay things out down to very fine resolution so you can get exactly what you need out of the sheet. Jay Bates has a nice video detailing his process for this in SketchUp.
For your solid wood, you'll want to come up with a total estimate of the amount of wood you'll need, sure. And you may want to estimate that assuming a couple different widths of the boards - 4, 5, and 6", for example. This allows you to quickly know how many pieces of wood you'll need to get at the wood store, as not every wood store always has 6" boards. Once you see the widths of the actual boards at the store, you can correlate that with your estimates. Colin Knecht of WoodWorkWeb explains this in a recent video.
If you have any special pieces in your project, or any unusually long/wide/big pieces, you'll probably want to list them separately. A single slab for a bookmatched tabletop; or thick, square pieces for legs; or two tall pieces for bookcase sides; etc. Those are pieces you want to make sure you can get out of one board and not have to glue two smaller pieces together. For other parts of your project, if you have to glue two or more pieces together it's usually not a problem.
You may wish to also keep a list of all of the pieces of your project which will be visible, so you can ensure you get wood with similar grain and color. For example, you can look at your list and determine you can get pieces A, C, and F out of this one board, and pieces B and D out of a second board. Non-visible pieces can be made out of the remainder, or with a board that doesn't have the same grain or color match.