For drawing multiple lines, you should first create a vtkPoints class that contains all the points, and then add in connectivity info for the points you would like connected into lines through either vtkPolyData or vtkUnstructuredGrid (which is your vtkDataSet class; a vtkDataSet class contains vtkPoints as well as the connectivity information for these points). Once your vtkDataSet is constructued, you can take the normal route to render it (mapper->actor->renderer...)
For example:
vtkPoints *pts = vtkPoints::New();
pts->InsertNextPoint(1,1,1);
...
pts->InsertNextPoint(5,5,5);
vtkPolyData *polydata = vtkPolyData::New();
polydata->Allocate();
vtkIdType connectivity[2];
connectivity[0] = 0;
connectivity[1] = 3;
polydata->InsertNextCell(VTK_LINE,2,connectivity); //Connects the first and fourth point we inserted into a line
vtkPolyDataMapper *mapper = vtkPolyDataMapper::New();
mapper->SetInput(polydata);
// And so on, need actor and renderer now
There are plenty of examples on the documentation site for all the classes
Here is vtkPoints : http://www.vtk.org/doc/release/5.4/html/a01250.html
If you click on the vtkPoints (Tests) link, you can see the tests associated with the class. It provides a bunch of different sample code.
Also, the vtk mailing list is probably going to be much more useful than stack overflow.