If good things come in small packages, then summer in Stockholm definitely qualifies. Even before the autumnal equinox, leaves are turning and mornings are cool.
Nonetheless, September may be the best time to visit as the cruise ships have departed and the crowds have thinned. Central Stockholm spreads across 14 islands whose bridges provide magnificent vistas of old world buildings and busy waterways. There are a variety of neighborhoods each with a distinct character. At the very least, make sure to spend time on Gamla Stan, the medieval core, exploring Södermalm (my favorite area) and Skeppsholmen. The city center comprising Norrmalm and Vasastaden has lovely pocket neighborhoods and parks as well.
Biking is a preferred mode of transportation in Stockholm. You can take advantage of the City Bikes, a system similar to those in Boston, New York, Paris and others, good for short rides, or rent one for a day or more. Visit Stockholm has excellent resources for biking as well as just about anything you need to know.
Be sure to block a few hours or even a day to enjoy the Djurgården with its endless bike paths, meandering canals and sylvan beauty. You will feel as though you have entered another world.