Remember that the gear itself adds stability, each rack unit is essentially a crossbrace. So there's no need to get too crazy with bracing if this will be sitting in your studio.
What I did on my racks is sink three screws per joint. You can also apply furniture glue to the joint when you put it together. If you want some extra bracing then a couple "L" brackets on the inside of each joint will do nicely.
The silver corner brackets are just ugly, and in my opinion are not necessary unless you will be using the rack on the road. If you will be gigging with this rack, then "all of the above" methods should apply.
The inside measurement of a rack is 19 1/8 inches wide. You see them from 19 1/8" to 19 1/4" wide. Again, that's the inside measurement. Each rack space is 1 3/4 inches tall. I've found it a good idea to add 1/8-1/4" to the top and a 1/8-1/4" extra to the bottom as well, for a little extra breathing room.
Cut the top and bottom boards to 19 1/8-1/4" wide and you are all set. Then add up how many rack spaces you need, multiply that times 1.75, add a quarter inch or so for breathing room, then add the thickness of the wood you are using for the top and bottom pieces (x2) and that will give you the height you need for the side boards.