Cat6 cabling is used for a 1Gbps network connection. It will still work with a 100 mbps connection.
Regarding the cable needed to connect the cable modem to your router, you could use any networking cable as long as you connect it to your "uplink" port on the router. Usually its the first or the last port on the router. (Ex. If the router has 24 ports, port 24 will be the crossover port.. and next to it there should be an uplink button you should push)