Cocktail Glass or Coupe
Also known as stemware, these will be your go-to glass for anything served up, with just a few minor exceptions. This includes both stirred and shaken cocktails. A standard coupe is a really classy way to serve most drinks, and you'll find that they are more versatile than the term "champagne coupe" would ever imply. It's hard to make a cocktail look bad when served in one of these. If you have extra space or want to make an investment, you could grab a set of nice Martini glasses, which are great for Martinis and Manhattans, or a nice v-shaped cocktail glass or Nick & Noras for a bit of variety. By forcing the drinker to hold the glass by the stem, you'll avoid having a chilled cocktail with no ice warm up too quickly.
Rocks Glass or Lowball
An All-Purpose Glass
If you've got the first two covered, the next best thing you can add to your collection is a glass to serve, well, everything else. This includes things that require specific serving vessels that you don't own yet(Tiki drinks, Moscow mules, Mint Juleps, etc.), and drinks that require a bunch of ice (Tom Collins, Bloody Mary, etc.). The first thing that comes to mind is nice Collins glass because they're super versatile, generally look good, and are easy to maintain. I've found myself using mason jars more than most, however, and they're usually easy to find close to home. If you're into the style you have an easy solution.