I finally have enough parts to start breadboarding! I bought a 3742-Contact Point Elenco Breadboard which houses four regular sized breadboards. I think I'll wind up needing two of them. Here's an animation of me testing out the 74LS283 adder chip, with inputs A1 and B1 being controlled by the buttons and Σ1 and Σ2 being wired to the LEDs. I've never done a massive breadboarding project like this before, so I anticipate being pretty slow at it.