I live in suburban Stockholm so here is a locals view on the topic.
Just taking a nice walk through Gamla Stan should be doable in a couple of hours so you will have plenty of more time to explore other parts of the city. In Gamla Stan - try to stay away from Västerlånggatan which is the most touristy street. Österlånggatan is a nicer option. The western part is a bit newer to try to stay east of Västerlånggatan for the oldest parts.
From Gamla Stan it is close to Riddarholmen with a couple of old houses and a nice view of Riddarfjärden - an inlet of Mälaren and the heights of Södermalm to the left and Norr Mälarstrand and the City Hall to the right. Riddarholmskyrkan is the burial church for much of swedish royalty.
Of the other parts I like Södermalm the best. There are nice walks along the water both to the west and east of Slussen. To the west there is a newly built pathway high above called Monteliusgången. To the east Fjällgatan may be good target for a walk. Further along the water is the high hill Fåfängan. A nice park to the southeast is Vitabergsparken.
In general the walks along waters are nice all over.
Here is a view from Monteliusgången shot by myself:
If you - like me - like national romantic architecture the City Hall is a must. It is possible to climb the tower. Nice because you get both great views and closeness to the brick. Churches in this style that stands out is Högalidskyrkan to the far west of Södermalm and Engelbrektskyrkan in the western part of Östermalm. The stadium built for the 1912 olympics is also worth a visit.
Other nice less developed islands are Skeppsholmen with remnants of one of the old naval centers of Sweden, Djurgården with several museums, cafes and nice areas in general and the old prison island Långholmen.
The nicest day trip is to just get on a boat to the archipelago. To get to the nicer outer parts in just a day will include spending several hours on a boat. The views will be nice but it may be a bit boring. The shortest trip is to Waxholm - it is a really nice summer city just a couple of hours from Stockholm by boat. You can save time by taking the bus in one of the directions. A nice island further out is Möja. For more of a partying vibe go to Sandhamn.
Other nice day trips
Nice smaller cities around Stockholm worthy of a day trip:
- Vaxholm was already mentioned - a nice summer city with a maritime feel
- Sigtuna is a cosy city on the northern beach of Mälaren with a couple of old church ruins and a nice old city center.
- Mariefred hosts Gripsholm Castle - one of the old royal castles of Sweden.