This step is crucial. I went with construction grade pine for 2 reasons. It is cheap and it is readily available at big box stores. It is a hassle to make something nice with construction grade wood, but it may be worth it if you're inexperienced (like me) and/or you're going to paint the thing anyways.
Spend as much time as you need to find the lumber in decent condition. Look for knots in the wood, damage from storage methods, and bows in the faces or the edges of the boards. The assembly can work some small bows out but you want straight wood.
My supports consist of the 2x2's joined together using dovetails. Dovetails are unnecessarily complicated for this task but I wanted practice. A dado joint would be perfectly sufficient. Check out our guides on both dovetail and dado joints and choose dado because dovetails are ridiculous for this task.
I didn't get a good picture of my process here, but this is the perfect use for the jigsaw. Take your time to mark out the board according to the schematics, then go slowly with the jigsaw. During this step, I had a scrap 2x2 handy to make sure my notches were the right size before moving on.
I made the mistake of pre-staining everything. This was a problem for 2 reasons. The stained wood was a mess and I had to touch up everything that got cut or notched.
I highly recommend doing a dry fit with your pieces, then paint or stain everything in one go. I won't cover which stains or paints to use here as that is a huge topic and one I'm not an authority on.
Placing all the uprights in the position I expected them to be, I started by sliding the bottom shelf through all the supports. Once it was close to the right position, I moved the supports in place and flattened the shelf. It took a little negotiation so I used a wood block on top of the shelf and hit that with a hammer to avoid damaging the wood. A rubber mallet would be ideal if you have one.