Ken's trip to Sweden!

Stockholm
The Old Town (with the Royal Palace) - http://en.wikipedia.org/wiki/Gamla_stan
Vasa Museum - http://en.wikipedia.org/wiki/Vasa_Museum
Skansen - http://en.wikipedia.org/wiki/Skansen
There are also tons of more museums, old churches, parks etc. to see.

Stockholm is a lot more expensive than most American cities so look carefully at the menu before choosing where to eat. Not sure about your ideas for staying the night but Af Chapman is one of the most charming "cheap" places so do not miss the opportunity unless you prefer a fancy hotel.
http://www.svenskaturistforeningen.se/en/Discover-Sweden/Facilities-and-activities/Stockholm/Vandrarhem/STF-Hostel-Stockholmaf-Chapman/

Copenhagen
You're gonna have to look this up, I've only been there once :)

Gothenburg
Bohus Fortress - http://en.wikipedia.org/wiki/Bohus_Fortress
Älvsborg fortress - http://en.wikipedia.org/wiki/%C3%84lvsborg_fortress
Feskekôrka - http://en.wikipedia.org/wiki/Feskek%C3%B4rka
Liseberg - http://en.wikipedia.org/wiki/Liseberg
House of the Emigrants - http://emigranternashus.se/?page_id=191
Shark Park ;) - http://www5.idrottonline.se/GoteborgsBF-BasebollSBF/English/

Gothenburg and the areas surrounding it has a lot of old fortress and such as it used to border both Denmark and Norway. It's also very centered around the fishing industry. Apart from this Gothenburg is almost as expensive as Stockholm and has one of the worlds highest rated amusement Parks.

Generally, visiting Sweden means seeing a lot of nature. Both the areas around Stockholm and Gothenburg has huge green areas and both also have their own archipelago. The are different from each other so in case you end up in Gothenburg too you should see both. Stockholm is almost fresh water while Gothenburg opens up towards the Atlantic. There are many many old buildings (castles, fortresses, churches) all over the countryside if that interests you.

Sweden also has the freedom to roam (http://en.wikipedia.org/wiki/Freedom_to_roam#Sweden) which gives you the right to walk pretty much anywhere you like except right outside people's house. You may already know about it, it's a fantastic opportunity.