To the Stockholm area
The main Stockholm airport is called Arlanda which services a wide range of international destinations. A smaller airport, called Bromma, offers connections to some Nordic airports. Other airports such as Skavsta and Västerås are a considerable distance away from Stockholm.
There exists a direct overnight train connection from Berlin and Hamburg to Stockholm. From Copenhagen there are frequent trains to Stockholm (5 hours).
From Arlanda airport
There exist several options to get to Stockholm from Arlanda airport. The airport train, Arlanda Express takes 20 minutes and costs about 300 SEK for a single, and slightly cheaper for a return. The airport coach, Flygbussarna, takes 45 minutes and costs 129 SEK for a single, and slightly cheaper for a return. Taxis should cost around 750 SEK.
Around Stockholm
The local transport company is called SL and operates a metro (tunnelbana), commuter trains, trams and busses, Every journey costs the same amount (42 SEK). You can pay directly with your bank card or phone if it supports contactless pay. You can also download the SL app which contains several options (e.g. 7 day pass). If the previous options do not work for you, you can buy a physical SL card which you can charge with a number of trips or passes. See https://sl.se/en/in-english for more information.
To and from the Royal Academy of Sciences
The address is
Lilla Frescativägen 4A
114 18 Stockholm
The nearest metro station is Universitetet (red line in the direction of Mörby centrum). You can also take a bus to Universitetet, e.g. bus 50.
As you exit the metro station, turn left and left and pass under the main road. Follow the road as it curves right. Once you reach a cross roads, continue straight taking the road marked "Vetenskapsakad.". Continue to the front of the main building and take the steps to get to the main entrance.