To split a polygon you probably want the 'Advanced Digitizing' toolbar. Right-click anywhere in your toolbars and a pop-up menu will appear. About half-way down you will find a check-box for Advanced Diogitizing. Click on that to turn it on. Once you have the toolbar you will find the split features tool.
To add points and lines, remember that a shapefile can only contain one type of geometry, which is why you can add polygons but not the other two. To create a new layer go to Layer->New Layer->New Shapefile Layer. Then you will be asked to specify the geometry type. Once you have a point or line file the Add Feature button in the Digitizing toolbar will add points or lines respectively.
Other tools you may find useful could include for this sort of work:
- Coordinate Capture
- Numerical Vertex Edit
- CAD Tools
There are loads of other plugins. These are just ones I use most often when editing vector data. To get these (and others) go to Plugins->Fetch Plugins. You can use Plugins->Manage Plugins to turn plugins on and off.