bourbon, rye, vodka, gin - all good to have around and can be your preference, plus generally good cocktail bases. rum and tequila if you are into those as well. i like to have a decent brandy on hand too
for mixers, you want citrus; lemons, limes, and oranges around for both the peel and juice, or just get those specific juices. tonic/club soda
red and white vermouth (sweet and dry). bitters (angostura and orange are good to start with). i like having sugar cubes around, and making simple syrup is easy to do at home.
below is probably good starter mixers that will make a wide range of cocktails, but really i'd just find some cocktails that you want to make at home and get the ingredients for those.
nonino
averna
cynar
aperol
fernet branca
branca menta
montenegro
campari