The game drops you in at the shallow end with a friendly tutorial stage. You are presented with the overarching story - essentially you are chief of your village and there is a Goblin horde on the loose.
At this point, you will be introduced to the Shop, which is accessed via the bottom right corner of the screen. You have to build a Cannon to combat the enemy threat. Click on the Shop and select the only item available, the Cannon, to continue. You will then be tasked with placing the Cannon, which you can drag around by touching the screen.
Clash of Clans features a relatively comprehensive tutorial
The game advises you to use Gems to speed up the construction of the Cannon. You should not do this, instead saving the gems for a future use as they are in extremely short supply.
You are able to freely move items around as much as you like, so don't worry about the Cannon's positioning for now.
Once the Cannon is ready, Goblins will attack. Watch how combat works and see the Cannon in action. You are also introduced to unit healing.
Next you get to experience the thrill of attacking the enemy. To place your units in the enemy village, tap repeatedly where you want them to appear. Bear in mind you want a fair spread, though there is no real danger of failure in the tutorial. Following this victory you will earn 500 Gold and 500 Elixir.
To decrease building times, the tutorial recommends you hire a second builder. To do this, tap the Shop icon and choose the 'Builder's Hut' for 250 Gems. Despite what we said about saving your Gems earlier, this is an essential purchase.
See off the Goblin threat
With that done, you are shown how Elixir works - build an Elixir Collector for 150 Gold to begin getting a steady supply of this important resource flowing in. Gems are not necessary to complete this. You should then construct Elixir Storage to house any overflow Elixir that can't fit in the Collector's capacity. Having completed that, you will probably level up.
The next task is to also increase the storage of your Gold. Repeat your task for the Elixir Storage with Gold Storage. You will then have to build a Barracks to combat the future Goblin threat. Tap the Barracks and place it on the landscape somewhere. Once it is built, tap on the Barracks and choose 'Train Troops'. You are only able to build the classic Barbarian units at this point. To do so, tap and hold on the Barbarian icon. Let the meter fill up until you have 20 in the queue. You can use Gems to speed up the wait, but it's a few minutes so shouldn't be too arduous.
During this period you will see the Barbarians spilling out of the Barracks and going to wait around the campfire. Once the 20 are completed, you will be able to attack using the icon in the bottom left corner. Pick the Goblin Forest from the map that appears and deploy your units as you did before.
With the battle finished and won, you will choose your name. Be sure to pick one that you are happy with as this is not something you can change later on. Following this, you should upgrade the Town Hall, as instructed. That concludes the proper tutorial section, and the entire screen opens up for you to use, including all the indicators and meters at the top and down the side. You are now free to do as you wish!