You need to take a bus to Drottingholm.
Certainly, but by metro you can get very close to Drottningholm. From Brommaplan metro station you can either take a short bus connection or walk. I find metro easier means of transport than buses in Stockholm.
Venturing to Falun or Engelsbeg seems a bit much. I would rather go to Visby, if you want to do a long excursion.
Well, it is not too much for me. Actually I think it is quite easy, at least easier than bus + ferry connection from Stockholm to Visby which could take minimum 4-5 hours.
- Train connections (about once per hour with a change in Västerås) from Stockholm to Ängelsberg takes 2-2,5 hours. From the Ängelsberg station it is a short walk or bus to Engelsberg bruk.
- For Falun, there are a plenty of options. 1) infrequent direct train (2,5 hours), 2) with train change in Gävle (3 hours), 3) train to Borlänge (2,5 hours) and change to bus (0,5 hour). The latter was what I did. There is plenty of time to explore the Great Copper Mountain (including the mine tour) with its environments and also the center of Falun.