Shaking is the most common way of mixing and diluting cocktails. If you need help choosing a set of shaking tins, check out this post. Once you have your tins, using them is pretty straightforward. The only difficult part for some people is getting them separated but with a bit of practice you'll be cracking them apart in no time.
To use your shaker setup, first place your large shaker tin on the bar1 and measure your ingredients into it. Add ice. Place your smaller "cheater" tin, opening down, into the larger tin. You don't want the vertical axes of the tins aligned when you do this. Offset the axis of cheater tin from that of the larger tin. Basically, you want the wall of the cheater tin to rest against the lip of the larger tin. Holding the larger tin firmly to the bar with one hand, slam your palm down on the bottom of the cheater tin to firmly wedge it in place.
Now, pick it up and, with the smaller tin pointed toward yourself, shake it. Pointing the smaller tin toward you makes it less likely that you will spray cocktail all over the bar or, worse, your guests, if something should go wrong and the tins fall apart mid shake.
I've read a lot of articles over the years about "proper" shaking methodology but trust me, if anyone has ever told you the "right way" to shake a drink, it's bullshit. I guarantee you given the exact same recipe, temperature, and dilution level, no one is going to be able to tell the difference between two cocktails, one shaken with a vigorous, piston-like, back and forth motion, and another shaken with a "Japanese hard shake". One method may dilute the drink faster, for sure, but as long as you know that your method is giving you the proper dilution, you could put your tins in a paint shaker for all the difference it makes.
My shake time changes depending on the cocktail but I basically shake until after condensation starts forming on the outside of the shaker tins. I'll generally shake longer for cocktails served up without ice than those served over ice because cocktails served up need to be to the exact right dilution level when they hit the serving glass. Cocktails served over ice will continue to dilute in the glass. With my technique and lots of Kold-Draft ice, this usually means about six to ten seconds of very hard shaking. Since I'm making 500-600 drinks on a busy night (around 60% of which are shaken), a brief, forceful shake serves me well. I have fairly large hands so I find it easy to shake one handed and, therefore, to shake two cocktails at once, but if you're not comfortable with that, lots of bartenders prefer a two-handed shake as well (it's probably better for your shoulders in the long run anyway). Do what works for you.
Not to get too off track here but I'd like to share a little bit of the science behind what's happening when you shake a cocktail. Once you slam that cheater tin down on the large tin, you've formed a closed system with a fixed volume. The liquid and gas inside the tins at this point are in isolation from the air outside the tin. As you shake, the ice starts to melt and cool everything down, the melted water has a higher density than ice (and therefore has a lower volume) and as the liquid ingredients (alcohol, citrus, sugar, etc.) cool down, they also increase in density (decreasing in volume). The volumetric decrease in liquid and solid ingredients allows the air in the shaker to increase in volume, but because the system is closed and no new air molecules can be added, increasing the volume of air decreases the air pressure inside the tins. The negative pressure forms a vacuum, drawing the two tins together. This is one of the reasons why the tins don't fly apart when a bartender shakes one-handed and why the tins are difficult to pull apart. The guys over at Cooking Issues, who I really respect, have a much more in-depth explanation if you're interested. They also have a fascinating breakdown on how shaking chills your cocktail below the freezing point of ice (around -7°C), even though you start with ice at 0°C and ingredients at room temperature.
Okay, so you've got your drink cold and properly diluted, now comes the "hard" part, breaking the tins apart. Holding the bottom shaker tin firmly in your non-dominant hand, place your thumb over the point where the cheater tin touches the large tin and, using the meaty part of your dominant hand where the thumb meats your palm, hit the large tin firmly at a spot about an inch and a half away from where the two tins meet. You should hear a crack and the tins should break apart. Practice with empty tins, or shaken ice water until you get it.
Once you have the tins apart, strain the drink into your serving glass, rinse your tins, and drink up!
1 Some references will tell you to measure your cocktail into the smaller tin instead of the larger tin, and then pack it with ice. This way, they observe, you will never fill your tins past their capacity. While it's true that there is a danger of overfilling the larger tin—making it impossible to get the smaller tin on without spraying cocktail everywhere—over time you'll learn what your maximum capacity is and you won't overfill. I prefer to measure into the larger tin for two reasons. First, the smaller tin can usually hold only two cocktails worth of ingredients max, whereas the larger tin can hold three easily with room to spare. Also, because the tins are already oriented properly (with the smaller tin facing up, and therefore, toward you) measuring into the larger tins allows you to grab them off the bar and immediately start shaking without having to flip them around. This especially comes in handy when you are shaking two cocktails at once.