Depends on what you want it to look like

We'd benefit from a better shot of the kitchen. Please tell us more about this space. It'll inform the advice. I'd paint the ceiling between the beams. I might also be tempted to whitewash the walls.

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I left my beams and ceiling dark. It matters how you decorate/paint under the ceilings which is important. Paint the cabinets white; paint the wall a light color (I chose a light green); add good lighting. There was no power in my dining room ceiling, so the electrician hid the box and wires behind a beam. It ran from a wall outlet, up, and all was covered with a cord cover.

It looks quite dark and heavy to me. How high is the ceiling? If it's about 8 feet my preference would be to paint the ceiling and beams a soft or creamy white. The ceiling in our family room is 8 feet, and the beams, ceiling and walls are all white. It was that way when we bought the house, but I would have done the same thing. It's definitely a personal preference, but sometimes dark beams can "pull" a low-ish ceiling down lower. If your ceiling's higher, you might be okay with just painting between the beams.

You may need to take a look at the overall room and decide your ultimate goal. Dong things piecemeal can lead to wasting money

You might need to lighten both the beams and the wood ceiling as your ceilings are pretty low.

I have wood ceilings, and they do majorly darken a room compared to white. white ceilings bounce the light down and act almost like an additional light source. ultimately I decided to not paint mine because we get tons of natural sunlight and the wood is lighter than yours (pine with a reddish tint).

on the other hand, your room looks super dark and that shade of brown seems drab. that might just be the photo you took. is there any way you can post more during the day, or maybe from other angles? based on that photo alone, it does seem like your space would be improved if you painted the ceilings.

Depends on what your overall plan for the space is. The oatmeal looking carpet will tend to look dingy if the ceiling is white. Changing one or two things won't solve the problem without taking all of the space and furnishings into consideration. What are your natural light sources? How many lamps do you have? Are you taking advantage of all the electrical outlets to upgrade the light in your space? Unifying the mix of surfaces which look to be white brick, orange-ish paneling, upper cabinets, island and ceiling, could improve things a lot. More pix will help.