How can I run an EXE from within my application?

Use the Process class found in the System.Diagnostics namespace.
[C#]
Process proc = new Process();

proc.StartInfo.FileName = @"Notepad.exe";
proc.StartInfo.Arguments = "";
proc.Start();

[VB.NET]
Dim proc As New Process()

proc.StartInfo.FileName = "Notepad.exe"
proc.StartInfo.Arguments = ""
proc.Start()