I have a modem from my cable company; I pay rent on it monthly.
To save money I bought a modem at a rummage sale; 2wire by Gateway.
It doesn't have a coaxial port, only telephone and Ethernet ports.

I want to connect the coaxial cable to the 2wire modem.
Is there a connector/splitter to go from coaxial to Ethernet?

It sounds like you bought a DSL only modem. They don't work with cable and therefore you cannot connect your coaxial cable connection to it. It would seem you bought a nice paperweight.