I had been to Sweden for few times. It is not very ideal to go in September because the summer is gone but the winter is yet to come. Anyway...
Stockholm is a great city. You should visit Gamla Stan. You can consider to start your trip from t-centralen, walking toward to the Gamla Stan and Slussen. It will take around half a day.
Cityhall is worth to visit. They should still have guided tour during weekend. You can check out more information at their visiting center homepage.
http://www.swedenvisitor.com/swedenvisitor/home/
About Museum, you can visit Vasa Museum. You can take a boat ride from Slussen to Skansen. Vasa museum is nearby. Skansen is a life museum also worth to visit. If you have the weekly pass I talked about below, the fee for the boat ride is included.
Transportation in Sweden is generally not very expensive, if you stay long enough. If you stay there for a week, you can go to 7-11 or PressBjorn (Spelling shold be wrong, a convenient store anyway...) to buy a weekly pass. It should be around SEK 250.
If you only stay for few days, you can consider a Stockholm pass which is around SEK 480. It includes all the entrance fees for all museums in Stockholm... If you visit the palace, cityhall, vasa and 2 more... it should cover the cost...
Dinning out in Stockholm is expensive. But lunch is OK. If you stay in apartment where you can cook, you can consider to go to supermarket, cook the meal and it is actually cheaper (i think) than in Singapore...
Going from Stockholm to Uppsala, you can take train. The ticket is not that expensive... for sure not 600 SEK.... it should be around SEK 100-200. And there are trains from Stockholm city to Uppsala every hour.
Uppsala is an university city, but it is quite small. One day should be enough. You can take an early train to Uppsala and return to Stockholm in the evening.
About Norway, Bergen is not very big. However, you should visit the fish market which only opens during weekend. Besides the fish market, there is a old town and it is also quite fun to visit. However, the main point to go to Norway is to visit the fjord... but taking a boat trip to see the fjord seems too cold...
It actually depends on how you plan the trip, you can also consider to go to Helsinkin. You can take an overnight boat to there...
Besides, there are quite a lot of Stockholm's pictures in my picasa album. You can visit...
But I did not upload the pictures in Bergen....
http://picasaweb.google.com.hk/matchz/