BUS: It is about 40 minutes away by bus if you take the 670 from Tekniska Hogskolan in the northern area of Stockholm city. Tekniska is a northern travel hub, it is along the red line and it is the beginning station (Ostra station) for the Roslagsbanan (commuter train to go north) and it is the first stop for many of the buses headed to the eastern islands of the county.
BOAT: Depending on the time of year there is a ferry (1453) available that will take you from Strömkajen in Stockholm to Vaxholm. There is an extra charge to use the service and it takes about an hour.
Tip from a local: Buy a period travel ticket, there is the 24 hr, 72 hr, 1 week, and 1 month. Depending on which one you buy, you will have unlimited travel, which if you are using the public transport a lot it will be cheaper. If not, then each ticket will be ~40 SEK/ 4 USD.