CDL,
I put together the following cocktail supplies list for a friend who wanted to start playing around with cocktails at home:
bitters - angostura, peychauds, orange
amaro - gran classico is a really good one, and very versatile
vermouths - white/red
liqueurs - maraschino, cointreau
mixers - tonic, club soda, ginger beer
optional - absinthe / benedictine (don't get these to start)
tools - shaker tins, mixing glass, hawthorne strainer, julep strainer, bar spoon, jigger
All of the tools can be ordered really cheap online/Amazon. For shakers, I prefer small tin/big tin as opposed to pint glass/tin.