I should clarify, it's not a dumb switch, but it is a managed switch that I just hadn't done any tinkering with. So it has been passing the VLANs just fine. I've since discovered that, as I imagined, my problem pinging PCs from my phone was Windows Firewall settings, which I've corrected. The problem pinging my NAS across subnets turned out to be a limitation of the phone app I was using. Tried a different app and I was connecting to the NAS no problem.
Thanks, everybody, for the help! Everything's up and running now.