This is a quite famous bike tour for bikers starting from Stockholm.
Getting to Täby can be quite tricky if you don’t know the place and you want to ride on bicycle paths. Anyway there are signs guiding you to Täby.
Passed Rybdo you need to cycle for some 4kms on a big road where cars drive pretty fast. There is no other way to get closer to Vaxholm from there but you can use the emergency lane when it is available and in Sweden car drivers are very kind to bikers.
Once out of the big road you can ride quietly for 10kms on a typical swedish road sorrounded by woods and with few cars passing. This road will take you to Vaxholm. There you can take a ferry which will bring you to the island of Rindö. The crossing takes about 5 minutes (time tables and other info here).
On the other side of Rindö another ferry will take you back on the “mainland”. From there you can enjoin other 15kms of quiet road which will take you close to Gustavsberg.
From Gustavsberg the way back to Stockholm is mainly on bicycle paths.