So, lets start up QuantumGIS. You should get a screen looking like this
In the upper right you have a list of the layers in your map, much like the layers in a graphics editor. In the centre is the view of the map, and around the edges are toolbars, which can be dragged to new positions as needed.
Now, we'll need some data to start with. A good source of free vector data is Natural Earth at http://www.naturalearthdata.com/
Click Get the Data. This gives you the option to download various kinds of data suitable for several different scales. For now, lets use the low resolution Small Scale data. Under cultural, download the Countries and Populated Places datasets, and under Physical, grab Land, Oceans, River/Lake Centrelines, and Lakes/Reservoirs.
These will download as ZIP files, open them and you should see 4 files all named the same thing except for the extension.
Each of these sets of 4 files is a shapefile and contains one particular feature set. Create a directory for your project and extract all of the zip files into it.
Now, back in QuantumGIS you can add these feature sets as layers. Use the Add Vector Layer button (Green square with squiggles and a plus symbol), click Browse, then in the open file dialogue, go to your project directory, and select all the files. Click open in both dialogues and you should see the layers appear in the layers list, and the features appear in the map. They will probably be strange colours and in the wrong order.
You can drag the layers up and down the list to get things the way you want them. To zoom in and out, use the mouse wheel or the zoom buttons (+ Magnifying glass, - Magnifying glass). Pan by middle dragging, or by selecting the pan tool (Hand) and left dragging. You can also zoom to the entire document (Magnifying glass with four red arrows), or to the currently selected layer (Magnifying glass and blue square)
You can also select features. The select icon is the yellow blob with the mouse cursor over top. Select it, and the layer you want to select things on, and you can click to pick the features you want. Hold control to add to your selection and click the icon with the yellow blob and the red "not" symbol to clear your selection. You can change layers and start a new selection without affecting the selection on the previous layer. You can also click the dropdown menu beside the selection icon to get freehand, polygon, and radius selection tools. Try playing with them. When you have a selection, you can Zoom to Selection (Magnifying glass and yellow blob)
Next to selection, there's the identify tool, which looks like a mouse cursor with an i on a blue dot. Use this and click on a feature (Again you have to have the layer selected). A box will pop up telling you all the information about that feature. Try selecting the countries layer and then clicking on a country.
There is also a measure tool (ruler) Like select, this has a drop down to select distance, area, or angle measurement. Just click several times to create the shape you want to measure. A dialogue box will appear to show you the measurement as you go. Right click to finish the line/polygon. Check the "Ellipsoidal" box to do the measurement on the Earth's surface, otherwise it's done on the flat map.
Next, you can look at a table of all the information about all the features in a layer. You can call up the table for the current layer with Open Attributes Table (Window with horizontal lines in it). Any selected features will be selected in the table, and you can adjust the selection by clicking on the numbers at the left side. You can sort the table by clicking on the column headings. This is useful for figuring out what data there is to work with, though it can still be a bit obscure at times. Good data should come with some documentation to explain what the different fields mean.
So, now to get rid of those ugly colours.
Double click on the ocean layer to bring up the layer in the list to bring up the Layer Properties, select the Style tab.
If you click the Change button on the left with the colour swatch, you can quickly change the colour.
If you want to get fancier, click the Change button with the wrench icon underneath the preview. This will bring up the Symbol Properties.
Right now, the layer has one layer in its symbol which is a "Simple Fill" just a fill with a solid border. You can adjust the size, colour, and style of both the fill and the border. This simple fill is probably just fine for the ocean so lets leave it and go to the Symbol Properties for the countries layer. The countries are blocking out the land underneath, so set the fill to None. Borders are often dotted, so try changing the border style as well. You can adjust the symbols for the other polygon and line layers the same way to get a map that looks more reasonable.
The point symbols for the cities let you pick shapes to use but are otherwise the same. Here's a city symbol made from two simple symbol layers.
Besides the simple symbols, there are other types you can use as layers in a symbol. You can make lines out of point symbols, points from a font glyph, or patterns from SVG images. If you are going to be editing the map in a graphics editor afterwards, you will probably want to stick to fairly simple symbols at this stage.
That's enough for now. Hopefully up next, symbols that vary with the attributes of the data, labels, projections, exporting.