Shake or Stir
The oldest argument at the bar, and what the evidence actually says.
Before anything else
Ingredients go into the shaker or mixing glass first, ice afterwards. Fill it about two-thirds. That looks like far too much, but a mean pour of ice melts almost immediately and you end up drinking the meltwater.
Use full-size cubes rather than crushed. Small ice has much more surface touching the drink, so it chills and dilutes very fast, and by the time everything is cold it has already gone watery. Bigger cubes take longer to do the same job, which is what gives you room to get it right.
Which method, and why
Shake anything containing juice, dairy, or egg white. The extra force aerates the drink and builds a thick, velvety froth that's as good to look at as it is to drink.
Stir anything made only of spirits, unless one of them is a cream liqueur. You're not chasing froth here, you want a smooth, strong consistency with minimal dilution. Some hold that shaking a drink with nothing to cushion the impact "bruises" the liquor and leaves a bitterness that stands out among more delicate flavors.
Never shake anything carbonated. You'll destroy the bubbles. It matters enough that the fizzy element should go in after the drink is strained into the glass, followed by the gentlest possible stir to integrate it.
The Bond question
The shaken-versus-stirred Martini argument has drawn in actual scientists. John Hayes, a food science professor at Penn State, has pointed out that shaking chips the ice into tiny flakes, which dilutes the drink and gives it a more watery character, though those same flakes keep it cold longer.
Bartender Darcy O'Neil tested the theory that Bond's preference came from shaking better incorporating the oils in potato vodka, and found that shaking does dissipate that oil. He agrees shaking makes a drink colder, but notes that after roughly twenty shakes a cocktail is about as cold as it's going to get.
Culinary scientist Dave Arnold agrees on the temperature, but adds a catch: extreme cold suppresses your ability to taste, sweetness most of all. His fix, if you're shaking something traditionally stirred, is a dash of simple syrup to compensate.
The verdict
Want the traditional experience with a spirits-only drink? Stir. Want the ingredients as thoroughly combined as possible? Shake. Want it colder? Shake. Want it to taste sweeter? Stir. The two methods produce genuinely different drinks, and which one is "better" is a matter of preference, not fact.
Working the shakers
A Boston shaker is two tins, and the greater interior space gives ingredients and ice more room to move and emulsify, which is why professionals tend to favor it. A Cobbler is the familiar three-piece with a built-in strainer: easier to handle, and some bartenders find it gives better control over the ice.
Boston method: ingredients in the smaller tin, larger tin two-thirds full of ice, invert it over the first without spilling, seal the two together, and shake hard for about twelve seconds. Tap the seal while holding the larger tin to break it, then strain through a Hawthorne.
Cobbler method: ingredients in, ice to two-thirds, cap on, shake vigorously for fifteen seconds, a little longer because there's less room inside. Pull the cap, hold it by the top, and strain.
Stirring takes two to three times as long as shaking to reach the same chill. Ingredients in, ice to two-thirds, then move the bar spoon in a smooth circular motion rather than up and down.