When I hook up a router to a SpeedTouch modem my internet connection stops working

This is likely being caused by an IP network conflict. SpeedTouch modems use the private IP space of 192.168.1.X to provide NAT/DHCP to devices on the LAN side of the modem. Some DSL routers also use the private ip space of 192.168.1.X on the LAN side. When you hook the two together the router now has an IP address of 192.168.1.X on the WAN side and also the LAN side, so the router ends up not knowing where to send the traffic. In order to remedy this situation you must change the LAN address and DHCP pool of your DSL router to something other than 192.168.1.X (use 192.168.14.X for example). Read the instructions delivered with your DSL router or contact the manufacturer of your DSL router for specific instructions on how to make the necessary changes.