Apple supply the TC without cables.. except power.. yeah!!
You need to go and buy a patch cable.. just a few dollars in any large computer store.
The TC is not a modem.. you MUST NOT plug phone into the TC.. please immediately unplug that.
The TC is a router.. you will need to keep your modem working as it is .. and most likely it is easier to just setup the TC as a bridge device.. and plug it into a LAN port of the existing router by ethernet to the WAN port (convenient but not required) of the TC.
You bridge the TC via the airport utility.
LaPastenague is correct in saying you need to setup the Time Capsule in bridge mode. It sounds like you are using a modem router combo.
Just get a regular ethernet cable(same as a "patch cable") and connect it from the LAN port on the modem to the WAN port on the Time Capsule. Go through the setup as normal and if it asks you to put it in bridge mode do so. (if you are using Mac OS X 10.6.8 or earlier it will prompt you for it. If you are using a Mac OS X 10.7.5 or later it will automatically put it into bridge mode if it is receiving a private ip address).
To verify bridge mode:
10.6.8 or Earlier: Airport Utility > Select Time capsule > Manaul Setup > Internet Icon Along the top > Internet Connection Tab > Connection Sharing. If it says "Share a Public IP" Switch it to Off(Bridge Mode)
10.7.5 or Later: Airport Utility > Click on the Time Capsule > In the window to the right of it click "Edit" > Network Tab > Router Mode. If it says "DHCP and NAT" switch it to Off(Bridge Mode)
Hope this simplifies things....