Turns out I really shouldn't have trusted Sports Basement to put the bike together. I figured "what could they possibly screw up?" Turns out, a lot of stuff.
First off, one of the shifter cables is catching on the stem bolts and is preventing the bike from steering properly. That's only potentially fatal.
Secondly, the cable guide in the bottom bracket is very clearly printed with "USE WITH METAL SUPPORT" and guess what's missing? The metal support.
Finally, the clueless retard managed to use a piece of brake cable housing to route the shifter cable to the rear derailleur. These are literally physically incompatible (brake cable housing is 5mm, shifter housing is 4mm) so you have to go out of your way to bodge it into working.
I'm going to be getting my money back on that service, and hopefully they'll be able to find the missing parts, although if they're gone for good I can just buy them online.