The wood is gorgeous, just too much. Prime and paint out the walls in a warm white, add quartz countertops in a mostly white colour( with maybe a fleck of colour or a little veining in it), new white subway tile and new hardware on the cupboards. Then a nice big indoor/ outdoor area rug in the dining area and maybe a runner or smaller matching one in the cupboard area( maybe a colour to tie into other parts of the house. Not to bright a rug, it’ll just add texture and definition and break up all that tile. Add a bit of personality with things on the counters or pics on the wall. You have a beautiful home. Lots of work, but minimal investment. If you think you are going to swap out the cupboards later or do a full gut reno someday, just choose inexpensive laminate counters for now, some beautiful options today.

The beams and ceiling seem to not have the same amber shellac or polyurethane finish based on the sheen of your photo. Those finishes tend to orange or yellow over time. This could be a blessing and make color correcting the tone of the wood a lot easier to get it more neutral or ashy.

Based on your inspiration photos, I'd check out the firm McAlpine House to see some of their fantastic designs that utilize similar ceilings, albeit newly constructed. You have the bones, so I'd personally utilize it.

I love it, Just add color by painting the soffit over the cabinets, and with bright colored mid century modern chairs, barstools, and accessories (placemats, area rug, small appliances.) Hopefully you like mcm.

This is a sweet little MCM.

I’m assuming the whole house is “dated” in a good way.

This house tour might be inspirational. They kept the paneling and all.

https://www.houzz.com/magazine/houzz-tour-a-family-home-comes-together-one-piece-at-a-time-stsetivw-vs~33122387

Ceiling looks like it matches the cabinets? If so I would start with painting the walls and the island white. You’d be amazed what painting that could do. Natural wood cabinets are making a come back.

This is certainly NOT a hot mess kitchen. I think it has a lot of potential. I'd embrace the wood paneling and the cabinets, which look to be good quality. My biggest problem is the flooring and the counter tops. I'd replace the counter tops with white quartz, change out the hardware and lighting, and replace the flooring with marmoleum, a very 50s type linoleum tiles. Your idea to paint the ceiling between the beams would look nice, but I wouldn't make the beams darker.

Here are some ideas:

Green Cottage kitchen · More Info

This homeowner went very bold to emphasize the 50s vibe of the home:

My Houzz: Bold Vintage Color in a 1962 Austin Ranch House · More Info

I think it'd be a mistake to paint the ceiling. I painted our covered porch pine ceiling after priming first. It was a bitch to paint and the knots eventually showed through. Your wood ceiling is beautiful. Leave it be.

Not a Pro.

I would just change out the counters and floors.

Too much orange wood

I also say “Don’t rush into any changes”. If you don’t already have an ideabook for your new home, start one. There are already some beautiful ideas among the comments here.
If the kitchen cabinetry is in good condition, plan on keeping it for a good year or more while you save more money and come up with a plan for the whole kitchen-dining room space and any other contiguous area not shown in your photos. Post additional photos to show us the rest of this space. During your planning time, have the electrical system inspected and updated if necessary. Clean or change the furnace filter, inspect the roof and plumbing and do any necessary repairs. This home has a lot of potential.

What a great space. If it were mine, I would put heavy linen-like curtain panels on the window wall from the sliders past the windows. I would put a great and bold light fixture wherever you’ll put the table (and get rid of the one that’s there). I would replace the kitchen cabinets with creamy white cabinets in a design that has a statement hood over the stove with a tile backsplash and new countertops in a materials that please you. I would extend the higher countertop so you could put some barstools there, again my choice would be covered in white linen, tho others would enjoy bold colors. Then set the table with a cream colored runner and great tabletop setting. As budget permits I would replace the floors with something more organic. Have fun with your new house!

Kitchen wood panels look very good.

what are the counter top and cabinets made of? If particle boards, I would replace them with granite and better quality cabinets. You can put more lighting like the ones in bars.

Here are a few ideas to play with. Photos from Houzz.

I would work with it for at least a year and try different colors and styles to see what you like. It's important to live in a space for a while before you can appreciate all it has to offer. Please do not paint anything. Get new counter top, cabinet hardware, lighting, and don't bring and natural wood furniture into the space. You need color, metal, glass, plastic.

Unless you are up for a gut job address the lighting and be done for now. Live with stuff as is for at least 3 months , that kitchen needs a redo for sure and you will find there are things you will want to change there . No dark beams with all that lighter wood, slow down painting the wood is not easily undone .

Until you have a full plan for the entire space, don't touch a thing. Anything you do on a $10K budget is only going to make everything else look really old and dated instead of "charming" as it stands now. The best thing you can do is take a year and put together a plan. During this time you can gather photos that are appropriate for the style of your new home ( I noted you have one Ideabook which if it is suppose to reflect what you hope to do with this house, I think this house isn't meant for you unless you have a very large budget), gather information and put together a realistic budget and save, save, save.

And please, whatever you do don't paint that ceiling. It really is the fabulous part of your home and with the right plan, it can be the cherry on top! Enjoy your new home.

Lean into the look, it's charming. Otherwise 10k will not go far and only make the kitchen look like a hot mess as you say, totally disjointed.

This kitchen is NOT a mess. Do not paint the wood ceiling. It is beautiful. People pay top dollar to add wood ceilings and beams. I also like the wood walls.

Agree with others to sit tight, save your money and gather ideas and inspiration for a new kitchen. $10,000 is no small sum. It can buy a good portion of a new kitchen when you are ready. No need to waste it on painting the ceiling. The ceiling is not the problem. McCarthy367 provided some great ideas for kitchen with warm wood ceilings such as yours. I like the last photo with the dark wood island.

In 2004 I changed the white laminate counters of my builder white laminate kitchen. I put in a nice green gray laminate counters for $1800. I did not want to spend $5000 on granite, which was so popular back then. It was OK, but I was never really happy with the color. But so glad I did not waste any more money.

Finally in 2018 after many years of planning and reading Gardenweb and Houzz, (thank you everyone, buehl, live wire oak, etc.) I replaced my kitchen with a white high gloss cabinets and quartz counters and I am so happy. So don’t waste your money, cause you will likely not like it and then you will not want to rip out the new work. Buy a new faucet and light fixture to tide you over while you plan. 🌸

I personally wouldn’t do much here, as it’s a nicely preserved retro kitchen. If you must paint, please don’t paint the ceiling. Maybe just the walls..white...

address the lighting, new chandelier for sure

maybe change counters, backsplash and floor, but stay authentic to the time period

I agree with most everybody. Do not paint the ceiling. It is very pretty. The wood paneling could be painted white, but it would take a far amount of prep work; cleaning, sanding, priming to get a good finish look. With all the warm wood tones I think you need some cool colors for balance.
I understand you want to change the kitchen to make it “your kitchen”. What about new stainless appliances?

I also love the ceiling and come down on the side of "don't paint it!"

I would probably remove or paint the wall paneling, as it seems much more orange than the ceiling and cabinets.

What condition are the cabinets in? If they're good, I would keep them. Updating the hardware could help modernize the whole look, and I would replace the counters with something light (probably quartz or marble, depending on your patina tolerance). You could look at one of the fairly inexpensive quartz patterns like Caesarstone Organic White.

If the cabinets aren't in good enough condition to keep you could look at replacing, probably with IKEA given your budget. Personally, though, I would rather devote the budget to painting the walls, getting a new counter, and if there's enough left, replacing the floor. If you paint the walls white, I would consider a hardwood floor.

If you replace the counter, also consider cutting the taller section of the counter down so the peninsula is a single height.

Good luck! This is definitely not a hot mess, and with some minor changes I think it could be gorgeous.