Best basic cocktail ever (but a non-starter if you don't have or like the ingredients) is a Negroni: 1 gin, 1 red vermouth; 1 campari, stir, serve over ice in a short glass with a sliver of orange peel or a slice of orange - if drinking before dinner this is perfect as an aperitif with little savoury nibbles, but if you want it as a wintry long drink you can top up with ginger ale (for afternoon drinking...)
I also like an old fashioned: bourbon, teaspoon (or less) gomme syrup, two dashes of orange bitters, again, stir and serve in a short tumbler.
New favourite for fresh and tasty mid-long drink is an adapted diabolo: 2 tequila, 1 liqueur, 1/2 fresh lime juice, top up with ginger beer (not ale). You can switch around the liqueur to what you like/have in the house. I like it with Cointreau, OH likes it with kir, DSD likes it with chambord.
Rum cocktails are trickier because I really like to disguise the taste of rum! Mojitos: 2 rum, 1-2 fresh lime, lots of sugar (Demerara or muscovado), lots of fresh mint, bash together soundly with end of the rolling pin, top up in long glass over ice with soda. Pina colada - massive throwback but had forgotten how delicious they are: white rum, coconut cream, pineapple juice in the blender with ice until it looks like a milkshake; keep adjusting the proportions until it tastes how you want it (the amount of coconut cream you get from a 400g tin of coconut milk is enough for about 3 cocktails, so about 6 measures rum and half a carton of pineapple juice). Serve with an umbrella (mandatory).