Quite good kit for the beginning comes with cocktail shaker, muddler, scoop, strainer, knife, and case. Different types of muddlers are not essential gear since you can use other tools in your kitchen, but if you choose kit of muddlers, be sure that they are not lacquered or glazed, because that coating wears off over time and could get into your drinks. A spoon will help you stir it up a bit better, and it doubles as a smaller measuring stick, pay more attention to long spoons since some cocktails require high cups.
Most cocktails require shaking or stirring, so you can pretty much get away with using only a cocktail shaker and long-handled bar spoon. Nearly every cocktail recipe that calls for shaking or stirring calls for straining so that the drinks are separated from using the ice in your shaker/stirring glass so you can serve the cocktails over-iced (or on top of large chunks of ice to keep the drinks from getting too watered down). If you are using a Boston-style shaker, you will want to pick up something called a Hawthorne Strainer so you make sure the ice that you used to chill your beverage does not get into the glass and dilute your cocktail.
If you are making a cocktail with the shaking method, then you are straining with a hawthorne strainer, mentioned above, then you will still have small pieces of ice, and potentially other bits and pieces, making their way into your cocktail. Strainers are perfect for when you are making cocktails and you want to avoid having tiny bits of ice or bits of fruit floating around in your final drinks. Strainers help to keep the small bits of ice within an empty metal mixing bowl, and outside the cocktail.