HDMI Connection

It's a Contour stream player. I don't know the model number.

Do you need a separate remote to switch between inputs?  I wonder if any switches are HDMI-CEC compatible.  If you have UHD equipment, make sure its UHD compatible.

These are just thoughts for the OP if he or she decides to go with a switch.

It's not UHD. I don't know what CEC means.

I thought this would be a good solution, but I see that I'd have to use a switch button. The ports are behind my TV, which is in a corner. If I'm going to have to move it to get to the switch, I may as well move it to change from the cable box to the DVD player. If I'm misunderstanding this, please let me know!

An HDMI Switcher isn't a switch you physically flip.  It's a device with multiple HDMI ports for your devices and one cable to connect to your TV.  It just adds more ports to your TV.

My question is how do you change or select a connected device...a dedicated remote control of the switch or can you pair the switch to your Contour remote.

HDMI-CEC.  As you know, you connect devices with cables.  These cables carry audio/video data.  However, HDMI-CEC can use these cables to carry communications.  Not your personal communications but inter-device communications.  For example, if you power on a Blu-ray player, the Blu-ray player sends a signal to the TV and the TV...through HDMI-CEC...automatically switches its input to the Blu-ray player.

An HDMI Switcher is a solution but is it worth getting one now if your TV is kinda old.  Could you move this TV to a bedroom, den, etc?

I'm not ready to replace the TV, which is already in the bedroom. As far as the switch, the ones I was looking at online did have an actual button to press. Maybe I'm not understanding this, so I'll probably order one and take my chances. In answer to your question about changing devices, I use a remote--a separate one for the DVD player--but I have no idea if the Switcher will work with my Contour remote. Maybe that's where the mysterious button comes in. . .