I think that you’ve identified most of the things already– learn about how logic circuits work, learn how nixie tubes work, and try to figure out how that particular clock works — or is supposed to. Learn how to use a multimeter, watch out for the high voltage, and see if the power supply is working, and then see if the logic circuits are working.
The electronics hobby clubs are mostly called “hackerspaces” and “makerspaces” nowadays. Go meet the people at TXRX and/or similar spaces around. Find one that you like and join it. (Meet the people, and learn _how to learn_ what you need to learn. Don’t ask them to fix it for you, or to teach you to fix it.) Go to the mini maker faire next year. There are lots of resources, and it sounds like you’d like to dive in.
