Pretty granite. Can you show us a picture that includes your cabinets? With what I see, I would look to match the lightest color in the granite and a simple tile like a subway. Your granite is the star.

The lightest color from the stone, which looks like a cream, in a simple subways tile. You want the backsplash to be as simple and understated as possible so the stone can shine.

The flooring in the adjoining living room is a light colored carpet. We are wanting to put new flooring throughout the kitchen, living room, etc. that is all the same and are thinking a luxury vinyl flooring as it seems very reasonable for families. That will be my next dilemma is what color of vinyl flooring that looks like wood would look the best with my cabinets.

Would a ceramic type subway tile or a natural stone subway tile be best?

I would suggest a cream textured tile, exactly what you said. 'not too busy'. This will make your granite pop and add some flair to your kitchen.

I have a cream subway tile with my oak cabinets:

https://www.houzz.com/photos/my-pics-work-in-progress-phvw-vp~57088087

I have a natural oak wood floor.

https://www.houzz.com/photos/my-pics-work-in-progress-phvw-vp~58830325

I wold bring home lots of flooring samples - look for warm browns if you are going for a wood look.Sometimes you just need to see with samples what works and what does not.

This is my "wood kitchens" idea book - might see something that inspires you:)