and connecting the SpeedTouch modem to that. You'll have to download the modem's firmware to the router and enter your username/password combination into its firmware so it (router) can make the connection each time it's booted.
Connect your computers to the router with normal network cable and RJ45 connectors and away you go - each machine will be able to use the connection without any of the others needing to be on.
Many routers now have an ADSL modem built in, so you can dispense with some of the configuration. You may prefer to buy one of those and be done with it.