Although i couldn't tell you the best place to dock, that would depend on when and how. Sweden is very boat "dense" and in the summer months, there may be few available slots.
There are plenty of varieties of charts for Swedish waters, for leisure boaters. "Batsportkort" is the most common variety. And the Stockholm archipelago is normally divided into north / middle / south in that "brand" of charts.
There are indeed three major routes, however, the southern route is not something i'd advise you to take because of the navigational risks.
My suggestion, would be to take the same route that the Stockholm / Riga ferries take, fairly wide channels most of the way and deep enough to handle large shipping.
For the "Batsportkort" choice, the "middle" chart will cover that approach.
That route will take you in just south of "Sandhamn" a popular stop for boaters and also the beginning and end of the "gotland runt" sailing regatta.
The last option is to use the north entry route, which is the one used by boats to and from Helsinki, this route requires the "north" chart. As well as the "middle" chart.
YOu can see the Stockholm - Riga path here on google maps, and use that one to determine what charts you would need.
Google Maps
The north chart covers north of "Moja" and the middle chart covers as far south as "Dalaro"
Hope that helps.