The old town of course (Gamla Stan) is a must see and is the oldest part of Stockholm and dates back from the 1200s. This is an area you should stroll around and see the Castle, and different squares. Eating ice cream from one of the waffle cone makers is something nice when it's hot out. On Gamla Stan
Two neighboring islands to Gamla Stan. Riddarholmen is perhaps a bit less interesting than the Parliament island and the old town, but it does have a boat dock there that offers a ferry to another part of Stockholm (Solna I believe). I'm going to be talking about other ferries and boat options further down. The walking street leads right through the middle of the parliament building structure. I've been on a tour and it's ok. Viewing the island in general, especially at night, is recommended.
If you were to keep walking up that same walking street, Drottninggatan, then you'll end up on Norrmalm. This is probably what
most people consider the center of Stockholm and is home the the main trainstation, the central subway station (which are close
but not exactly in the same place), and even has a nickname in this area as: city.
This is a big square next to and beneath a major part of stockholm's best shopping. There's a large glass statue that many people like. I think it's hideously ugly :) The best I can say about this part of Stockholm is that it connects to other places that are more interesting. On Sergels torg
These are the kings garden and are situated directly across the water from the royal castle located in Gamla Stan. Kungsträdgården
This is the ritzy part of Stockholm and is an island home to a lot of famous hotels, buildings, and even one famous road.
Royal theater... Just a pretty building Royal theater
This is road I mentioned above that starts from Raoul Wallenbergs Square and makes its way to the bridge that leads to Djurgården.
This is the so called animal island is itself a big park with a nice trail that will take across the whole island. I'll point out a fiew choice places worth visiting on this island. Djurgården
This is an absolute must see. It's a museum of a very famous warship that sank jsut moments into its maiden voyage. The ship has been preserved in a very humid building and is a truly unique visit. Some info here
Outdoor zoo/museum housing nordic animals. It's not quite like a normal zoo where there animals are in cages and imported from afar. This is a place swedes would kill me for not mentioning. It's also just a classic swedish attraction. Skansen
This is our version of Six flags Gröna lund
Kungsholmen is another island. It's west of the train station and is home to a great many restaurants and cafes located both on the shore and on boats moored to the shoreline. Lots of eateries and pubs that I like are available on this island as well.
Situated on Kungsholmen this is always translated as "City hall" which implies it's a house of government and it's not. This is where the Nobel dinners are held. It's a worth a visit and from here you can talk a ferry to Drottningholm which is the official residence of Sweden's royal family. This is a great boatride and one i've taken myself and in contrast to several of the others it goes into Mälaren (the freshwater lake to the west of Stockholm) as opposed to the Archipelago.
The aforementioned castle... A super pleasant place to visit. Some info here
And a link to the ferry website that can get you there
Stockholm's archipelago is a UNESCO world heritage site with some 30000 islands and is home to vibrant communities, fortresses, sea pubs and restaurants and some unique galleries. Info
These are the closest islands that are considered part of the Archipelago and you can get to them by a ferry from nybrokajen. Some info here
Another island in the archipelago. We like this island because it has a lot of places to eat, pretty views on the way there and an impressive fortress across the water (there's another electric cable ferry that you can use to get there from Waxholm). There's a fantastic cafe there on the wather overlooking the sound where all the boats come in. A bit on the fortress itself Kastellet