Copenhagen & Stockholm

I don't have any useful advice, since I am going to Copenhagen next week for the first time, but in case you haven't already looked at these, here are Rick Steves' videos on Copenhagen and Stockholm:

https://www.ricksteves.com/watch-read-listen/video/tv-show/copenhagen

https://www.ricksteves.com/watch-read-listen/video/tv-show/stockholm

Do look into the Copenhagen Card and the Stockholm Card. They can be good values if you're a busy sightseer; they include transit and admission to most things. The Copenhagen Card is good not only for the city, but the whole region, so it's particularly useful. As always, you have to do the math to make sure it is right for you. But I was struck in Stockholm by how many of the Stockholm Card users were Swedes - who knew a good deal for that potentially expensive city.

Speaking of which, I hope you have Rick Steves Scandinavia - he has lots of good budget tips for these places.

I really liked the Millesgården in Stockholm, not in the center but worth the trip if you like sculpture. I enjoyed the National Museum too; the Vasa was worthwhile but crowded, but the nearby National Museum was empty. My boatride to the Drottningholm Palace was nice, as was the palace itself (particularly the old theater). But if you do this, don't make my mistake - use sunscreen!

Overall, I just enjoyed the vibe of Stockholm, and found walking around more of a "must see" than most attractions.

http://www.visitcopenhagen.com/copenhagen/transportation/tickets-prices
Four and more bus rides in 24 hours and you are ahead. We got by without 24 hour passes with two bus rides and walking around for each day. Copenhagen is walk-able and seeing the town the slow way is enjoyable.

24 hour pass may be convenient for you in not having to have coins. But note that the drivers will return change so exact coinage is not necessary.

You can (need to) buy train tickets at CPH in the terminal (staffed window or machine). No conductor on the train and you would be fined if found aboard without a ticket. Note: Central Copenhagen is zone 1, whereas the airport is in zone 4. The zone numbers get higher the further away you get from the city centre. Most other destination are only one or two zones.
https://community.ricksteves.com/travel-forum/denmark/from-the-airport-to-downtown-copenhagen

CPH to central station = 36 dkk Local bus in zone = 24 dkk

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Thanks for all your tips, our trip went wonderfully, first time we didn't get a car in Europe
London - I know it well, we stayed at the novotel paddington and used the tube via oyster cards to get everywhere..London Eye, British Museum, Trafalgar Square, Tower of London, Buckingham Palace, Westminster Abbey

Copenhagen - train from airport to central station was very easy, stayed 2 minutes from the station at Axel Guldmusson, loved walking around Copenhagen and definitely recommend hop-on bus to get a feel for the city (although there are competing companies with the same color buses so it can be a little confusing)
round tower, rosenburg palace, amalienburg palace, meat packing area for new restaurants, Nyhavn area.

Stockholm - bus from airport to central station - easy to find and get on, stayed at Radisson Blu Waterfront, awesome hotel and great rooms and perks with business class rooms. We again used hop-on bus to learn the layout of the city. Stockholm was great but not as great as London and Copenhagen, lots of construction everywhere plus they have made some decisions about modern buildings in the main part of town that takes away from the traditional beauty of more classic architecture.....but overall a great great trip.

Thanks you for the questions and follow-up. Planning an unexpected trip to both those cities in late August.

In Copenhagen, we enjoyed the canal boat tour. I think there may be more than one route available.