Every kind of food processing has trade offs including cooking which destroys vitamin C among other things.
White rice was historically missing thiamine (B1) found in brown/hole which was killing off a surprisingly large number of people in Asia as it became popular. Now days foods are often fortified with various things to cover the most common and severe issues, but that’s probably not enough on its own.
In terms of highly processed foods you run into various additives. There’s a long list of minimally studied additives/dyes/preservatives/etc being used only for them to later be recalled and replaced with some other poorly studied substances. Ignorance may be a legal defense for Nabisco adding X to their products, but it’s not going to protect your health.
So, the advice is basically to have the absolute minimum amount of processing steps which is a surprisingly useful heuristic alongside having a varied diet including lots of different plants.
Industrial processing. As a simple rule, the longer the list of ingredients on the food label, the more processed. The ingredients aren't something you'd normally think of as food? More processed.
Another way I've heard it stated: "single ingredient foods." And as you prepare them, try to keep them that way.
For example, adding a ton of butter to potatoes makes them 10x less healthy. More or less, calorically they become fries. Fries taste great but they should be an occasional treat, not a staple.
Keep in mind that if you buy a bunch of "let's pretend it's not processed" "healthy" foods and use them to cook a meal, you're a) creating an ultra-processed meal, and b) it now has a long list of ingredients, you're just not writing it down.
I take it you interpret "save money" to mean keep yourself barely fed, clothed and housed, and otherwise consuming the bare minimum so that you can maximize the amount you put away?
If not, you might want to loosen your interpretation of words in other domains, too.
Unless I run out in the field and bite a cow, that meat is going to be processed.
Likewise with potatoes, I'm going to be eating unpalatable starchy uncooked potatoes.