What style are you envisioning for the kitchen?

Instead of cosmetic changes that are akin to putting lipstick on a pig, rethink priorities.
Even if cabinets were structurally sound, they are hopelessly dated and nonfunctional. Horizontal pulls. No drawers to speak of. Laminate cabinets? Probably on particleboard substrate. No fridge surround.

No amount of quartz or granite will make cabinets look better. If anything it will make them look worse. A family member was hellbent on getting granite countertops and new tile floor in her late 70s kitchen. Spent her budget on two bath remodels, tile and granite. Then the base cabinets collapsed from weight.

PRIORITIES
1. Delay new appliances unless dead.
2. Forget quartz or granite for now.
3. Take measurements and consult with a kitchen designer to get best layout.
4. If cabinet footprint changes, ensure your refinished floor will cover areas now hidden by cabinets.
5. Get new quality cabinets. If all cabinets are shown in pic, price should not be bad. Worst case: get only base cabinets and get uppers later.
6. Get a Wilsonart HD prefabricated countertop without the 4" backsplash. Quartz on old cabinets would have been thrown out with the cabinets. Foregoing quartz will pay for a good amount of cabinets. Wilsonart about $20/SF.
7. Paint and lights.
8. New appliances as you need and can afford them.

I have to be honest, I am envisioning granite countertops while keeping your remaining cabinets and hardware. I am also suggesting you keep those counter stools, they are unique and add character to the room. I would also suggest getting stainless steel appliances and you'd be surprised what you can find. I know Sears offers appliance kits (stove, refrigerator, and microwave) for around $2,000. You might also consider changing out those closet/washer-dryer doors.

As far as actually being able to support the new stone, the only true way to tell would to have a professional come to your home and see the current condition/quality of your existing cabinets.

Our old kitchen was dated, non-functional and small. Had to do in three phases, still need to add MW, backsplash and some trim.

Window location remained the same. All we kept from old kitchen was the faucet. Not shown are two 34" wide pantries with 5 ROTs each recessed into a former double closet.

Replacing counter tops and appliances is more than a facelift. Counters could run you $75.00 a linear foot; good appliances at least $13,000.00; refinishing floors is a BIG ticket; new lighting and hardware adds more to your bill.

I think, before you take any advice from us, you go and price out your options.

You may find you need to keep what you have, or rip it all out and begin anew.

Not with updating to that 'large of magnitude. A small tide you over like paint, until you rip it out, sure. If you are going to do that much, you're over the line. Gut it. Start over.

Price out your options, sure it can cost a fortune, but do your research and get estimates you might be surprised.

I too am redoing an 80's melamine kitchen ! Depending on your budget...I would replace the doors and drawer fronts with a shaker style and paint them a more modern tone. Replace the hardware. Look in to a formica countertop..they have many beautiful patterns now..some that look like granite. Hope this helps.