Start by creating a new project.  You can only have one project open at a time, but each project can have many different compositions (mini projects) within it.

You start by importing files.  Do this by doubling clicking in the project panel.  Start off by importing an image file.

Now you need to create your first composition.  Everything needs to be in a composition before you can animate it or do anything else with it.  Click the New Composition Icon at the bottom of the project panel to create a new composition.  Then drag your image into the composition panel.

Click on the drop down arrow beside Transform in the Timeline panel.

You can change values for any of these options by clicking on the numbers on the right and dragging left or right (or you can type in new values)

If you want to change a value over time, click on the stopwatch to the left of the name.

Change the value, then drag the timeline indicator along a bit and change the opacity value again.  You will see little diamonds appear in the timeline which show where the values have changed.  pressing Spacebar will play your composition and you can see the opacity change over time.  Experiment with the other Transform values to see their effect.

Adding Effects

Look in the Effects and presets panel on the right.  Browse through some of the effects. When you find one that sounds interesting, drag and drop it onto your composition.  If it is an animated effect, press Space Bar to see it animate.  (Note that some more complex effects may take some time to render and play, depending on the power of the machine you are working on)

You can see the effects you have added in the Effect controls panel (which sits with the project panel)

Clicking the fx symbol next to the effect name enables you to see what it would look like if you took the effect away.

If you want to get rid of an effect completely, simply click on its name, and press the delete key.

Browse Presets

After Effects has a nice option that enables you to browse preset animation effects.  Choose Animation > Browse Presets.  There is a very extensive library of effects.  You can see a preview in a pane on the right.  When you find an effect that you like, double click it to apply it to your current composition.