Probably the best cocktail shaker is the boston style shaker. It's easy to use, can be used to stir and shake, and easily fits a bar strainer.
There's also the Cobbler shaker. This one comes with a built in strainer, but they tend to be sticky after getting cold and don't hold as much liquor as a boston style shaker. I've spilt more than my fair share of cocktails with these, so be warned.
I guess it depends on what you, personally, want to go with.
Out of all the equipment here, the jigger is the only MUST have, even for home use. Accurate measurements will go a really long way in making sure your cocktails taste as they should! Personally, I recommend getting one jigger that has multiple measurement lines inside of it, especially if you're using it at home.
This is only really necessary if you intend to use a Boston style shaker. They're used to strain cocktails so ice and pulp don't end up in your drink.
Used to mix and layer drinks, really. They're long, so they can fit in tall glasses or shakers. Unless you wanna make some really fancy cocktails though, you'll probably only use this to stir.
BAM BAM BAMMMMM ME SMAAASH!
Sorry, had to get that outta my system.
Muddlers are used to mash various fruits, herbs, and spices at the bottom of a glass. The most famous drink that requires a muddler is the Mojito, as you have to muddle the mint.
If you already drink wine, then chances are you have a corkscrew. It's not really necessary unless you plan to open bottles that have a cork inside.
Cutting Board & Knives
You probably already have this. You'll just need it for garnishes and cutting fruit for juicing, really. Pretty easy stuff really.
The most peculiar item on this list, a zester is used to grate zest from lemons and whatnot. Again, this isn't too necessary. You can totally skip over this unless you care about the pizzazz!
Chances are you already have this too. It's used in some cocktails, but not all. The most famous example is likely a Pina Colada, which must be blended. Please remember that most recipes must be blended until smooth, I've made some rather chunky cocktails on accident by not blending long enough before, and that isn't too pleasant.
This is just a juicer, really. Some recipes call for fresh juice, which you need to make with a reamer. If you already have a juicer though, you likely won't need this.
Sort of necessary, but if you have steady hands and pour slowly, it's not entirely needed.
It's pretty useful at times though, especially if you want very precise measurements and don't wanna risk potentially dumping the whole bottle of liquor on your counter.
Basically just a smaller straw used to drink cocktails. You don't really need this, especially if you already take it slow while drinking cocktails. (I highly recommend that btw, some sweet cocktails don't hit you until you've already had about 3 or 4)
Used for garnishing, basically. I've already touched up on this before, hence why it's totally optional. There are also swizzle sticks that fulfill the same function. Although you can get some pretty neat cocktail toothpicks, such as little swords!
I'm quite fond of the cool ones (///￣ ￣///)
Eh, these are mainly used to stir cocktails. Again, this isnt too necessary. However! There do exist flavored swizzle sticks that you can use (usually in the form of rock candy)! So if that's in your interest, go right ahead and get them.
Phoo ok. So glasses, yes? These aren't entirely optional, I'm afraid. The reason I'm putting them in the optional section is because there are like 20 different glasses to use. Realistically, you only need a couple, unless you're working at a bar or about to host a really fancy party.
So long as what you have can fit drinks, then you should be alright. Some recipes that require topping might be better with the proper glass though, so keep that in mind.
In the future, I'll add a new page to the site dedicated to different types of glasses!
This stuff is for if you're in a really tight pinch and can't really afford to get bartending equipment. This won't be a guide on how to find cheap shakers, jiggers, whatever; rather, it will be a list of things you may already have at home that you can use as a substitute for regular bartending equipment. This stuff won't do as good a job as regular bartending equipment; however, you're mixing drinks for yourself like a chad! Who cares if it's not Vegas casino bar levels of quality?!
Reusable Water Bottle
If you don't have a cocktail shaker and you want to shake things up a little bit, go ahead and grab a reusable water bottle. You may want to test the bottle to see if it's good for shaking first though; from my experiences, certain reusable water bottles will leak if you shake things around in them. Pour some water in first and shake that like you would a cocktail for awhile to see if it leaks, and if it doesn't you're good to go!
Coffee Filter/Fine Strainer
Fine strainers come in various sizes, but you might have one lying around the house somewhere. If you don't, hopefully you have a coffee filter. Both will be a little wonky to figure out, but unless you want pulp in your cocktails then you'll want to use some method of straining. I wish I had more advice on the matter but... I don't (个_个)
The chefs out here are going to cut off my head for even suggesting using liquids in a measuring spoon, but please hold your horses here. If you can't afford a jigger and you need some way to measure out liquids for your cocktails, and you don't want to get wasted by tripling up your recipe to fit into a measuring cup's units, go ahead and try using measuring spoons. There's a good chance you have them in the house, and so long as you can steadily pour liquids into there, it should work. There will likely be more spill over than usual though, just a forewarning.
To help you out, here's a small list of conversions for you:
8 oz = 1 cup
4 oz = 1/2 cup
2 oz = 1/4 cup
1 oz = 2 tablespoons
1/2 oz = 1 tablespoon
1/4 oz = 1/2 tablespoon