The cabinets on the 70's AS products (including Argosy) are tricky and lightweight, and seemingly custom made for each unit they are installed in. They are not easy to take apart, especially to take a "few pieces" out.
Mostly you will find rivets holding them together, and the rivets are behind the vinyl wood grained trim which finishes the aluminum horizontal and vertical pieces. The tops and bottoms are generally joined with little L brackets and screws into the floor and the countertops. Interior panels are often held to the metal parts with barbed clips which were fit over the panel and then pressed into a slot in the metal. They can be a real pain to remove as they are a one way design, in only, no easy way to pull them out.
In my cabinet repairs and removals, I usually started thinking I could remove a "few pieces" but wound up taking the entire cabinets and countertops out too, and then re building them to re install them. I usually found so much loose stuff after I got them apart I was glad that I had done the total removal and replacement. The shake and rattle and roll of the AS over the years often had caused sags, level issues, pulled out screws and so on which were good to replace.
On the countertop trim. If you remove the trim insert you will find that most of it was either power stapled to the plywood countertop, or power nailed with little brad like nails. You must remove each one carefully to take the trim off. If you are installing new laminate on top of the old generally there is enough overlap so you can move the trim up the 1/16 inch needed to re install it. New trim insert can be made from thin strips of laminate, or vertical venetian blind slats, cut down in width to fit the specific size slot you are working with.