I currently am renting a router from TWC. My bill says that I am being charged $5.99/month for internet modem lease and another $5.95 for home wifi. If I buy a Motorola SBG6580, will I be able to save on both those charges because this model is a modem & WiFi?
Yes, but buying a standalone modem and standalone router is recommended over a combo unit like that. Better performance and you can upgrade more easily as your needs change.
Buying your own modem and router or a combo unit will make you stop getting charged. If I were you, these are the steps I would take:
1) Purchase your own router and your own modem separately.
2) Tell TW you only want their modem and would like a new unit without wifi (they may try to tell you to keep your combo unit with wifi disabled, but try to push them to send a new modem-only unit anyway).
3) Activate their modem with your router and return their combo unit.
4) Call them again (or chat) and activate your modem. Do not return the new modem they gave you until they warn that you will be charged for it.
The reason you want to do this is if you ever have a problem with your connection, they may try to blame it on your modem. If they do, try basic rebooting/troubleshooting with your modem. If that doesn't work, have them temporarily activate their modem (that you didn't return) and see if the issue goes away. If it does, then its time for you to buy a new modem or get it replaced under warranty.