I have a hard time getting sugar to fully dissolve when I'm stirring a cocktail (for example, an old fashioned). I'm using granulated sugar (https://www.dominosugar.com/sugar/granulated-sugar) and I tend to stir a small amount in a shaker with 100+ quick circles and the sugar still hovers in the middle and sits on the bottom without dissolving. What am I doing wrong?
Most cocktails use sugar syrup (e.g. simple syrup with a water to sugar ratio of 1:1 or 1:2) instead of granulated sugar. This eliminates the need to dissolve the grains in - typically cold - liquids.
When you consider powdered (confectioner's) sugar because of the smaller grain size, remember that they will most likely contain anti-caking agents like starch or tricalcium phosphate. I'm not entirely sure how much this would influence the final taste, though.
You should use 'superfine' sugar, which is broken down much smaller so that it'll dissolve better in cold liquids.
You can make your own by putting some sugar into a food processor and whizzing it around for a bit.
You can also make a simple or heavy syrup, so you don't have to worry about sugar dissolving. Heavy syrup will keep longer in the fridge, as the sugar in large amounts is a preservative
Joe and Stephie's recommendations of simple syrup and superfine sugar are the best way to go in my opinion, but if you're lazy, in a hurry, or don't have superfine sugar, you can make a "poor man's" simple syrup by kickstarting it with hot water.
For instance, if following a mojito recipe that calls for 1 oz of simple syrup -- which is normally made with equal parts by volume of sugar and water -- I'd mix 1/2 oz of sugar with an 1/8 oz of hot water, and then once mostly dissolved, add the remaining 3/8 oz of cold water. If making several drinks, scale accordingly, or just do this mixing in each glass.
(On a side note, you can allegedly make superfine sugar by processing granulated sugar in a food processor; however, I've seen others online complaints that it ruined the plastic on their processors, so take this suggestion with a grain of salt/sugar.)