For air bound travel you can choose Arlanda Airport or Bromma Airport (mainly national flights).
You can choose to take a taxi directly from the airport (Arlanda or Bromma). The prices vary depending on time of day but with Taxi Stockholm it should be around 750 SEK from Arlanda and 550-600 SEK from Bromma. Ask the driver when you enter to get a fixed price.
From Arlanda you can take a bus (Flygbussarna) or the Arlanda Express train to Stockholm Central Station. Bus takes 45 minutes, train, 18 min. Bus costs approximately ½ price (return tickets are cheaper). From there you can take a taxi (with Taxi Stockholm it costs around 550 SEK). Ask the driver when you enter to get a fixed price.
Public transport from Stockholm Central Station (T-centralen) to Skogshem&Wijk: Subway line 13 (red) to Ropsten, from there you take bus 204 towards Elfvik. To find the bus station, continue to walk in the direction of the subway train, then go down. To the right there are several bus stops in different rows. 204 is the last in the second row (marked with E). The bus stop for Skogshem&Wijk is called ”Furutorp”. There is an 8-10 minutes walk from the bus stop to the hotel. Total travel time, including the walk, is around 45 minutes. Tickets can be bought by swiping your credit card when entering the subway or at ticket machines. The tickets are valid 75 minutes and can be used on the bus to Elfvik.
The address is Skogshem&Wijk, Hustegavägen 1, 181 90 Lidingö
Link to Arlanda express: https://www.arlandaexpress.com/?lang
Link to buses and subway in Stockholm: https://sl.se/en/in-english