The aim of stirring is to mix, chill and dilute.
You can use a mixing glass or a mixing tin. The glass looks much nicer but has a higher thermal mass than the tin. Either chill your mixing glass in a freezer first or use a room temperature mixing tin.
My preferred mixing vessel is affordable and unbreakable: an 18-ounce metal shaker commonly referred to as a tin, even though they are almost all made of stainless steel. I favor metal because it has a much lower specific heat than glass. It takes less energy to cool or heat a gram of stainless steel than a gram of glass. Most glass mixing cups are thicker, and weigh much more than the average 18-ounce metal tin, so you can see that glass mixing cups represent a significant thermal mass—one that will affect the temperature and dilution of your drink. If you were to stir two drinks, one in a chilled glass mixing cup and one in a room-temperature glass mixing cup, there would be a noticeable difference between the two.
— Dave Arnold, Liquid Intelligence: The Art and Science of the Perfect Cocktail
Add all your ingredients to your mixing vessel and add plenty of ice. The size of the ice will effect stir time—smaller ice has a greater surface area which means your drink will chill and dilute quicker. The size of the ice doesn’t matter greatly but using ice that is too small will result in overdilution if you’re not careful. If you have to use shitty service station ice, strain off all the excess water first. I use roughly 1” cubes.
…simply push the bar spoon back and forth with your lower fingers while the top of the spoon rotates in the pocket between your thumb and pointer finger. Your hand never swirls around. Focus on always pushing the spoon against the inside wall of the glass—if the back of the spoon is in contact with the glass, the spoon will turn when you push.
Temperature is an important ingredient in your drinks, and colder isn’t necessarily better. Almost all stirred drinks are in their prime between −5°C (23°F) and −1°C (30°F), while some drinks can stand being warmer and some colder.
— Dave Arnold
How long you need to stir for depends greatly on your stirring speed and ice. If using a mixing tin, it should be frosted. Start with 30-40 revolutions, taste and adjust as necessary. Use a thermometer if you have one while you get used to stirring times.