Grapefruit is a great flavor to work with for both professional bartenders and amateur mixologists. The boring truth is that most flavors behind the bar are either tart (think any citrus juice) or sweet (most cordials.) Some whiskey is a bit smoky or oakey, but otherwise it’s basically sweet. Grapefruit joins a few other ingredients (like tonic water, Campari, and other bitter liqueurs) in offering something different.
To start simply, I would juice a grapefruit and make a cocktail with the result. A Sea Breeze is a great option if you already have grapefruit juice around; if you want to just mix the fresh grapefruit juice with vodka, we’d call that a Greyhound. (It’s a Salty Dog with a salt rim, but I think that’s probably more of a summer cocktail.) Campari and grapefruit juice would be good as well, or the botanicals in gin would mix nicely with the bitter citrus flavors in the grapefruit.
As far as infusing a liquor, I think a grapefruit-infused vodka could be nice. I wouldn’t use the entire fruit, just the zest. (If you want, you can zest the grapefruit before you juice it, so that you’re using the entire fruit.) You’ll want to collect the zest in long pieces to make the whole operation work and look better; for a reminder of how to do that, read this piece on twists. (If needed, you can use a vegetable peeler instead of a zester.) You can infuse the liquor with the zests for a few days in a covered container, then funnel it back into the bottle; or, since the zests don’t take up much volume in the bottle, you can thread the long twists directly into your bottle of vodka and let it set until you finish the bottle. It looks nice, it’s a conversation-starter, and it will remind you that the bottle holds more than plain liquor.