I'm logging on to the net>
My modem kicks in and starts to dial
The noises start
Dialing
connecting
Verifying etc etc
Is there any way I can stop the noise from the modem?

Turn off the sound. Or, get DSL or whatever those high-speed connections are.

If you have Windows XP, go to Start, control Panel, Network Connections, then right click your connection and choose properties. On the General tab, you should see your modem listed. Click the Configure box, and you should see a box that says Enable Modem speaker. Clear that box, and close. That should silence your modem.