Simple mixology cheat sheet.
A "measure" (sometimes referred to as a "part") is just one unit of whatever measurement you're using. Typically this is 1oz, which is 1 normal sized shot glass, but if you're making a couple of drinks at once, your measure might be different.
A "bar spoon" (I abbreviate this "bs") is slightly smaller than a teaspoon, so if you don't have one, use a tsp filled about 3/4ths or 4/5ths.
Generally speaking, drinks that are "clear" - i.e. they don't have things like citrus juice in them (Old Fashioned, Sazerac, etc.) - are stirred, while "cloudy" drinks (e.g. with milk, citrus, egg whites, etc.) are shaken. This matters more for cloudy drinks though, and you can often use either of the methods interchangeably.
Drinks are usually served "straight up" (aka "up") chilled & strained into a glass, "on the rocks" chilled & poured over ice, or "neat" not chilled but simply poured directly into the glass (e.g. good Scotch).
If a drink is shaken over ice, it's often served up, and sometimes on the rocks. If it's stirred with ice, it's usually served on the rocks. If it's spirits right from the bottle, neat (or some people prefer on the rocks for this also). But ultimately your preference is the correct choice. Personally I prefer spirits neat, martinis/vespers/balmorals up, and anything else on the rocks.
You can use any booze you like, though I tend to group them into three categories: low, med, and high quality.
Low grade liquor is often not worth using, but can be sufficient in a pinch (brands like Stolichnaya, Martini Rossi, and Cinzano).
Medium quality liquor is good for drinks, but often not quite good enough to drink straight (brands like Old Overholt, Smirnoff, and Bulleit).
High quality liquor is often too good (or expensive) to waste in a mixed drink, and best drank straight (good scotch, Belvedere vodka, Old Raj gin).
Making good, clear, long lasting ice is a difficult, complex subject, but there are some shortcuts that can get you decent quality without spending all day.
Simple syrup is just sugar water. It's very easy to make, and you can adjust the ratios to get the desired viscosity & sweetness you prefer. The TLDR instructions are: put sugar & water in a pot, cook for a while. A more detailed version follows:
Any "granulated" (common) sugar works fine, but my favorites are Demerara or Turbinado (both brown). These produce a slightly malty, rich flavor compared to white sugar.
Start with equal parts sugar & water - something like 1/4 cup sugar & 1/4 cup water. This will make a medium-thick syrup.
Just because. It can also be fun to add this to cocktails, and you likely have everything you need to make it.
I love these, and feel they need their own section.
Aperitivo & digestivo are drinks - typically made with wine and/or herbs - meant to help stimulate your appetite or help you digest (or both). Most can be drank by themselves, though they lend themselves very well to cocktails too.
The most well known category is Amari. Most amaro are cousins to vermouth, and have a richer, deeper flavor. They can be used along side or in place of vermouth as well. Most cost around $20/liter on average. There are many, many types of amari, but some of my favorites include:
Now for the fun part. My personal type/brand choices for booze are italicized next to the name of the spirit. Remember bs = bar spoon here. Links to some ingredients on Amazon at the bottom.