I was trying to clean refaced cabinets (I think done in the 70's or 80's) that have a substantial amount of grease/dirt on them. While cleaning with water and mild soap, the finish is coming off. Is there anything I can do to keep this from happening? Is there anything I can do to fix the areas where the finish has come off? Thanks!
It looks like the finish is sufficiently worn that you're not going to clean it without further damage. That leaves you a few options:
Replace the cabinets. That's obviously the most expensive option but you get new cabinets in a style you like. You'd probably want to replace the counter top at the same time. Fortunately high pressure laminate (e.g., Formica) is pretty cheap.
Refinish the cabinets, assuming they are in good shape otherwise. Staining requires stripping the finish back to bare wood and quite a bit of sanding. Then stain and top coat. Frankly it's a lot of work and you would still have the same style of cabinet.
Refacing where the doors and drawer fronts are replaced and face frame and other exposed surfaces are covered with a veneer of some sort. Generally an advanced DIY project.
Paint the cabinets. Rust-Oleum makes a "transformation" kit for just this purpose. Or you can use a suitable "regular" paint. Both are going to require some prep work but it's still a very doable project for a DIYer.