8 DAys to Scandanavia Ideas

I prefer to fly into Copenhagen--a great city. Then fly a budget air carrier over to Oslo (1 hr.) We took the Norway in a Nutshell train/fast ferry ride to Bergen--a wonderful smaller city. Then we flew home through Oslo from Bergen.
With only 8 days, I don't know that it's worth trying to see Copenhagen, Stockholm & Oslo. I'd say pick two.

It really depends when (month) and what do you want to see / experience.

So many options what you can do in these countries.

There's an easy train from Copenhagen to Stockholm. You can also train from Stockholm to Oslo. Or there is an overnight train from Oslo to Stockholm, so maybe one going the opposite way? So you have several options. You could fly into Copenhagen and out of Stockholm and enjoy the scenery on the train. But, that will take up the better part of a couple of days if you don't use a night train...while getting a feel for the landscape. A flight will remove you from your vacation, but you will have more time just at your destination. Only you can decide which you'd prefer. I'd prefer the former so I see more of the countryside. But your time is short, so weigh your pros and cons.

Rather than a budget airline from CPH, consider booking a multi-city flight. For example New York to CPH, then CPH to Stockholm, train to Oslo then return OSL to New York. We have flown SAS a number of times with a connection hub through CPH. The added cost of a stop over in Copenhagen can be insignificant.

And yes, try to fit in the Norway in a Nutshell. You could do the long day Oslo-Myrdal-Flam-Gudvangen-Voss Bergen. Then fly back from Bergen to New York via CPH. https://www.norwaynutshell.com/original-tour/#

BTW 3 countries in 8 days is a sprint. I would pick two.

An 8-day trip would be seven nights, correct? Not counting the overnight flight to get to Europe? You could fly into Stockholm and spend the first 2 nights there. Then take the express train to Copenhagen, which takes 5 hours. The train ticket can be quite inexpensive if you buy in advance. Spend 2 nights in Copenhagen, then take an overnight ferry to Oslo. The DFDS ferry will give you both your transportation to the next city and your "hotel room" for the night. Again, book the ferry way in advance to get the best choices of cabin and best rates. The ferry leaves Copenhagen at about 4:30 p.m. and gets to Oslo at about 9:30 a.m. Then spend your final two nights in Oslo and fly home from there. Because your time in Scandinavia is so short, consider taking a guided tour in each of the three cities. There's a company called Nova Fairy Tales that offers small group public tours (walking tours and bike tours) in each of those cities.

If you want to make the trip from Copenhagen to Oslo part of the experience rather than just transport, consider going by water: https://www.dfds.com/en. You leave Copenhagen in the afternoon, have a leisurely dinner onboard, a dance in the night club, sleep in your cabin (with toilet and shower), and arrive in Oslo in the morning.

Three countries in 8 days is doable but means rushing it, I would urge you two focus on two. And for eight days I think Stockholm and Copenhagen is the best choice, but it depends on what you are interested in and what you want to see.

SAS has direct flights from New York to all Scandinavian capitals, so I would book an open jaw ticket from them. Fly to Stockholm, spend some time there, take the train to Copenhagen, then after a few days fly back to New York from Copenhagen.

We have been to Denmark three times, Norway twice and Sweden once.
Best way to see Norway is a cruise, especially one that goes to the Arctic Circle.

I would skip Oslo. Fly into Stockholm, stay 3 days and take the train to Copenhagen and spend the rest of your time there.
You won't need a car. Be sure to visit the Vasa Museum in Stockholm. Find a hotel in Gamla Stan. In Denmark, take a free walking tour, then visit the Royal Palace and some of the castles as well as Nyhaven (take a canal boat tour). Also, take a tour that goes to the Viking Museum and some of the outlying castles.