Should I plug my ethernet cable in to my cable box or tv?

Hi, TexHogFan1971. If your service is DIRECTV STREAM, you can connect the Gemini Device to your wall through an ethernet cable. Visit https://www.directv.com/support/article/000091381 for more info. John, DIRECTV Community Specialist

You can't use a "splitter" to share one ethernet connect to two devices. You need an Ethernet Switch.  If you want your TV to have internet access, and don't want to use its WiFi connection, then you could use a switch to feed both the TV and the DirecTV Stream box. 

If it has multiple ethernet output ports, then yes. 

It won't be applicable if it's a splitter for coaxial cable since it's only designed for DIRECTV via Satellite cable boxes. If referring to an ethernet switch splitter, you can plug in your TV and Gemini Device, especially if you're changing inputs to use your TV's app. John, DIRECTV Community Specialist