My wife and I are in the midst of the same type of project, and had some of the same concerns.
Our kitchen is from the 70s, and we got pricing on new cabinets from a custom cabinet shop and also a bigger kitchen specialist that would provide and install cabinets, but not in custom sizes.
For our tiny kitchen, if we did nothing ourselves, we would be at around $10k for new cabinets, stone countertops, sink, flooring, and backsplash. The cabinets were the bulk of that cost ($7k-$8k), and they weren't even all wood construction. The cost just goes up if you want all wood construction, dovetail drawer joints, etc. We looked at Ikea, but their options for white cabinets only came in 1 or 2 styles. They are a good option for the price, but we couldn't get past the "plastic-like" finish on the most traditional white door style. I would suggest looking at them in person before making the decision. They may have exactly what you're looking for!
We decided to order new drawers and doors (including hinges and slides) through the custom shop (our existing cabinets are custom sizes) which will be custom painted and installed for around $125 per opening. We will refinish the boxes ourselves, in place. This decision saved us thousands, but we sacrificed gaining additional storage.
Our quartz countertop and undermount sink will be templated and then installed for around $1700, and we only have about 25 SF of countertop. Our other quote was almost $1000 more. Sometimes you can find pieces of granite or quartz on craigslist for crazy cheap, but you have to haul it away (on an A-frame) and find someone to template it and cut it, assuming it isn't the exact size you need. If you are OK with laminate, you can save some serious bucks there.
We're going to do a subway tile backsplash and probably LVT flooring...replacing the trim and door/window casing and doing some painting ourselves and maybe doing the flooring ourselves. It's looking like it's going to be around $6k for everything.
Hope this helps.