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I stuck a blackcurrant twig in the ground and this, the second year, it has fruited a small amount. It was more of an experiment and I don't really want it where it is - if it's safe to dig it up, what is the best way to go about it?

It should transplant easily at that age.
Dig it up with as much soil attached to the roots as possible and just plant it in a suitable sized hole. You need to do it when it is dormant, so any time after the leaves have fallen off, about late November, until about late February.
Don't do it when the soil is frozen though.

Or you could put cuttings in the "right" place and have several bushes.