How should I pay for small things in Copenhagen and Stockholm?

Just withdraw some money from the ATM, or exchange some at a bank/exchange bureau (ATM gets better rates).
They still use cash for small purchases, they aren't totally cash free. Some vendors still prefer cash for small purchases because debit cards, credit cards and also the Swish app (a payment app) charge fees to merchants.
Currencies aren't the same. Denmark uses the Danish Krone, Sweden the Swedish Krone. I doubt merchants would accept either interchangeably, and in any case you wouldn't gain in the exchange.

I visited both cities last summer. Credit cards are commonly used even for small purchases - you could use them for your street food and groceries and no one would bat an eye. But they're not "cash free," so there are plenty of ATMs around to withdraw a small amount of currency to use, if it makes you feel better.

I did encounter a couple of cafes and restaurants in Copenhagen that didn't accept credit cards. Some places in Christiania, especially, accept only cash, and there were a couple of others that accept only cash or pay-by-phone payments. So it was helpful to have some cash in those cases.

I pay for everything with card and I rarely even have cash on me. For the average cashier, cards is going to be a lot easier on them, rather than a tourist who don't know which coins are worth what and have to fiddle with change. The only place I can think of where cash is needed is if you pay for a single ride ticket in a bus or maybe the odd hotdog vendor.
And yes Sweden and Denmark have different currencies.

I spent three weeks in Scandinavia in 2016 and could have used credit cards everywhere with two exceptions. I needed change for public toilets and to use the coatroom lockers in a few museums. Other than that, no cash needed.

I think each time you use the CC you are charged a fee but I'm not sure if that is true with every CC and every time you use it. Anybody know?

Scandinavia including Denmark are technological and economic advanced societies and electronic payments including credit cards seem to be the norm. We paid for bus fare in Copenhagen using coins but credit cards were usable for most transactions.

Denmark uses Danish Krone (DKK) and Sweden uses Swedish Krona (SEK) which are not interchangeable, If you add Norway to your visit you would need Norwegian Krone (NOK) and if Finland you would need Euros (EUR). If you are one who wants to land with local currency in Scandinavia you would need DKK, SEK, NOK and EUR. Or you can just wait until in country and and do what Rick Steves recommends, get local Kroner appropriate to each of the Nordic countries using the in country ATM cash machines.

I withdraw money from an ATM as soon as I arrive and use that cash throughout my trip. I only use my credit card unless I have a bigger purchase, only because my debit/credit card charges a cross border and currency conversation fee every time I use my card. Although those charges may be around $0.50, I'm sure it can add up if you use your card for food, museums, hotels, souvenirs, etc.