… and you’ll continue to have lots to use. Mint grows like a weed. If it didn’t smell and taste good, we’d consider it a weed. It’s tireless. Please pull it out of your garden now unless you’re OK with pulling up unwanted patches and cutting it back several times a year.
That’s, of course, good news if you do like mint, because it means you can cut as much of the plant away as you’d like for your cocktails without worrying about damaging it much. Most cocktails that do call for mint call for either a short sprig, or for 5-6 mint leaves separated from the sprig. (Since the plant grows so rapidly, you can cut an additional sprig to use as a garnish for any mint cocktail.) Snipping off full sprigs does have the added benefit of helping to control the plant’s growth — once a stalk is cut at its end, it can’t grow any more. If your mint plant is still establishing itself, the opposite is true; pulling leaves off and allowing the stalk to continue expanding is the better plan.
On to the cocktails: the mojito is de rigueur, and is a good jumping-off point. Similar to the mojito is the Southside, which swaps out gin for the rum and (this point is often missed) lemon for the lime. Even simpler than the mojito or Southside is the julep, which is simply bourbon, sugar, and mint, on ice. (Unless it isn’t bourbon.)
The tequila mockingbird usually uses creme de menthe, but you can make it with fresh mint by muddling 5-6 leaves of mint with a lime wedge, then making the recipe as printed.
If you want to get a bit more generic, there are a whole family of older drinks called smashes that are basically prepared the same as a modern julep preparation. This is where you can have the most fun. Muddle a few sprigs of mint with some sugar and a splash of water, then add ice cubes and 2 oz. of any liquor you have in the house. Garnish it with whatever fruit you want and give it a try.
(By the way, the original difference between a smash and a julep, according to what I can infer from Jerry Thomas’ 1887 bartending handbook, was that in a julep, the mint isn’t actually muddled; the sprig is simply pressed against the glass to express the oil, then inserted basically intact as a garnish immediately before serving the drink.)