+1 +1 +1 on buying an old bike, tearing it apart, putting it back together.
Working on old bikes can be an exercise in patience. Be willing to tighten down a locknut eight or ten or a dozen times to find the 'sweet spot' on a hub adjustment. Be willing to wait a few days for penetrating oil to penetrate. Make friends with a propane torch, IMO a vastly underrated tool.
Ask questions. This is where the co-op comes in, and the better co-ops have 'Women Only' nights, so you can learn how you're doing it wrong without feeling like some man is telling you you're doing it wrong
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Be patient. Really, it's the most useful tool in your kit.