The single most important thing to remember when making a cocktail is balance. There are many dimensions to this rule though.
First, the flavors need to be balanced. Not too sour, not too sweet, and not too bitter (but there are many cocktails that lean towards more bitter than not). The ingredients also have to come together in harmony, similar to how it is in cooking. This is the most difficult one to accomplish, especially when trying to craft your own cocktails. But as long as you can tell that there's something off (even if you can't tell what's missing) that's a good first step. If it's too sour, add a little bit of sweet. If it's too sweet, shake your cocktail more or add more sour. Keep playing around until you get the flavor that you are going for.
Second, the temperature and dilution need to be just right. These two go hand in hand, and it will vary drink by drink. We want our cocktails to be as cold as possible, but if you shake for too long, your drink will become more watered down. The size of the ice cube also matters. Bigger ice cubes are generally used for drinks like an old fashioned because they have less surface area. The ice melts slower but keeps your drink colder. Spirit forward drinks require more dilution than say a sour. The best way to judge temperature and dilution is by taking samples (with a straw) after you think it's done and shaking/stirring more if necessary.
And lastly, appearance. You don't want a muddy looking drink, and prettier colors make people more intrigued by your cocktail. Mixology is an art form. Every last detail from the glass to the intricate garnishes will affect how curated your drink looks.