You should have:
- a cocktail shaker
- a muddler
- a strainer
- a measuring device, with 1/2 and 1 1/2 oz at a minimum
- a bar spoon (kind of optional)
- a reamer
I call a bar spoon optional, because any decent stirring device will do; but a bar spoon is also about the same amount as a dash, so when you're starting out and you're unsure how much a dash is, fill a bar spoon.
I'm told my shaker is "fancy", because it's heavy grade stainless and glass. OK, whatever. It's fine. Don't spend a lot on one (I think mine was a wedding gift). Get a strainer unless you're just completely badass at using the shaker as a strainer. I'm not good at all at that.
I have a reamer for citrus. You will, of course, use pre-made citrus sometimes. No one keeps 20 lemons around at all times, and individually squeezes each one, when they're making drinks at a party. Get the ready-made stuff when there's 20 thirsty drunks in your living room. But for fuck's sake when you're at home, or making a couple servings, c'mon. Use fresh citrus. I don't like those ones that are vertical, they have extra ... stuff. This one is small and light and fits in my bar drawer.
I had a muddler that was a stainless steel rod with some kind of melee-weapon head. It looked impressive but it sucked. The best muddler is simple. You can use a stick, provided it's clean. Don't get crazy.
Obviously you want 1/2 and 1 1/2 ounce measures, because about 98% of recipes use those, instead of the more correct part/ratio/proportional method. That said, 1 1/2 ounces is a perfect "part" measure.