If you have a DSL router, there will be an RJ11 jack to hook up to the phone line. That's how you connect to the Internet. You can use regular telephone cord to connect to the phone jack. The other ports on your DSL router are for hooking up your computer, printers, etc. Those can use CAT 5e cables.
If you have phones connected to the phone line, be sure to use the small filters that should have come with the DSL router. The phone will be much quieter with them in place. Don't use the filter on the line you are using for the DSL router. That will only stop the signal from going out the the Internet.