<<< If I buy a wireless router, does this mean I can connect the modem to the router...>>>
No, Tiscali will supply you with a USB modem which cannot be used with a router. You need to buy a combined "Wireless Router/ADSL Modem" and discard the Tiscali modem.
<<< then the Xbox, my dads PC and my PC all to the router to share the connection? >>>
Yes, these all connect to the router with either a wired or a wireless connection.