Visual Studio 2010 provides a check box at the bottom of the start page that allows the option to automatically close the start page when a solution is loaded.
However, Visual Studio 2008 does not make it quite as easy. One solution is to open the Macro Editor (Alt + F11) and add the following code inside of MyMacros -> EnvironmentEvents:
Private Sub SolutionEvents_Opened() Handles SolutionEvents.Opened
If DTE.ActiveWindow.Caption = "Start Page" Then
DTE.ActiveWindow.Close()
End If
End Sub