The cabinet color is not so bad. I would keep the cabinets the same and add hardware. As far as the countertop I would look at quartz or granite that has some light warm tones in it. Taking the countertop to all one level would make the space bigger & more useful. I would also paint the walls if you keep the cabinets the same. Gray isn't really working. Since it looks like you don't get much light I would go with an off white/beige.

Does the general layout of the kitchen work for you? Are you happy with it? Could it be improved? How many people live in the house and how many use the kitchen? You seem to have a fair amount of times stored above the cabinets -- do these items need better storage?

Are you planning on staying in this for a while? Are you updating the kitchen in preparation for a sale in the short term?

in order that you can keep your dark brown furniture...paint your living room a white of some sort. It looks dark there, and gray and brown are not great together anyway. Once you freshen up those walls, and enjoy the space, some kitchen ideas may come to you. I would leave the unusual counter, use the kitchen, think about some knobs/handles and just enjoy your new home. 6 or 8 mo from now will give you new ideas of what you may wish to do.....your appliances look fine....don't rush into anything now. You have a working kitchen.

New countertop, sink and backsplash would be a great improvement. It's not going to look new and pleasing to the eye unless you get rid of everything on the counters, especially all the stuff on top.