Other than not having a backsplash, I dont' see what's so bad about it.

You can easily do a subway tile backspash in the cream color in your granite.

Take the valance down, and get some curtains in your favorite color. You also need a rug under your table.

Lose the valence. Find a fabric for the cushions you love and pick a color for the walls from that. You will have to harmonize with the granite. Most people would say grey for the walls. sigh. in five years everyone will be repainting all that grey. Can you move the light fixture to center over the table and get something wonderful? Spend some real time and money adjusting the lighting. Go to a lighting store with these pictures and see what they suggest. That granite, which we can't see very well, will control your choices.

Are the cabinets in reasonably good condition? If not, then just paint. Understand that once the Backsplash goes on it is close to impossible to successfully lift the countertops off and replace the cabinetry.
I would suggest you establish a budget for the paint and tile, then go to a tile store and see what matches your budget. To keep costs under control, I recommend using a cost effective field tile with a 'deco' band of more expensive tiles such as glass. If you go to our Houzz page, you will find a public idea book of tile options to give you some install options.

Paint to get rid of what ever that is on the wall, change the fridge to stainless and personally I would leave the backsplash you have or else rip it out and install glass subway tile stacked ,up to bottom of cabinets to go with your granite. Make sure the grout is the same color as tile. Take down valance over patio door, change out lighting to track for better light in kitchen with the least amount of cost. Don't hang stuff on side of cabinets and if you want a rug in front of sink find a funny one to get away from gramma's.

Your kitchen looks nice, maybe just lacking some color you can bring in from an adjoining room? One thing I would do is see if your fridge door and be switched to swing to the left instead of the right. This can be accomplished on many fridges with just a screwdriver. Look for the little buttons that hide the mounting holes. It will be so much easier to use.

What does one call that piece of countertop material that stands upright, against the wall?

The back splash.

LOL, Molly, that's what I call it, too! Don't quite understand why the OP'r and others (incl. yourself) talk about needing one when it's clearly shown in the pics.

Change the paint to a more modern color. Maybe a shade of grey. Then do not hang oven mits on the sides of the cabinets (gives it the grandma's kitchen look). Take down the valance on the slider. Get rid of the chair rail and beadboard on the bottom of the wall. I belive the hardware, light fixture, and table set look fine. I would not do a tile above the granite backsplash. The kitchen is not that big and it could get too busy.

These look like photos of the previous owners' choices. You can probably reverse the fridge door handles. Pick some art you like, take some colors from that for new paint. Put some art over the sink or a cut-to-fit mirror or just continue the backsplash up. Change the lighting. Politely suggest that the sellers might want to take their valence with them to their new place (and any other window coverings that you intend to replace immediately).