I personally think that Stockholm is not the best city to visit in Europe, but one of the best to live in. This does not mean you won't enjoy visiting Stockholm for a short time. In comparison to cities in Italy and France, Stockholm definitely falls short in terms of culture, history and food. I recommend very strongly (as someone coming from a tropical country) to visit Sweden only in July and August. These two months you can rely on good weather - good in the sense that it will be sunny and warm (above 20 degree Celsius), though again, in the evening the temperature can drop again to around 15C, so it is just best to bring a jacket everyday no matter the season. All other months are basically winter to me - bleak, cloudy, windy and cold (around 10 degree Celsius average). It does not rain as much in Stockholm, but it can be quite windy. I think this is why IKEA thrives in Sweden, because people put so much emphasis decorating their homes, making every part of their home cosy. Indoor is where Swedes spend most of their time in.
A Swedish friend once told me, "A good Swedish girl always brings a jacket with her everyday".
Stockholm in the summer is one of the most beautiful places to be. You'll see the Swedes swimming in the lakes/seas and sunbathing on rocks like seals or any open grass in front of their apartment buildings.
Some information that may be helpful on your trip:
Get an SL card: http://sl.se/en/
SL is Stockholm's public transport card that you can use for trains, trams, buses and ferries. Very useful if you want to travel much around instead of relying by foot. Tunnelbana / T-bana = subway, pendeltåg = commuter train, spårvagn = tram, färja = ferry, buss = bus.
Drottninggatan
Drottning means queen, gatan means street. Drottninggatan (the "queen's street") is the main shopping street in Stockholm. It's a long street from Hötorget area all the way to Gamla Stan (the Old Town). If you need to buy toiletries, medicine without prescriptions, etc: find any "apotek" or pharmacist. If you want to buy Swedish food, go to Hemköp, a large supermarket found at the basement of Åhlens department store. This is all within reach from T-centralen subway station.
Slussen
Slussen is the gateway to the hip area of Stockholm, i.e. Södermalm. The view from Slussen overlooking Gamla Stan and all the waters around is always nice.
Panoramic view of Stockholm
The best panoramic view of Stockholm (and, free of charge) can be captured along Katarinavägen (near Slussen), Monteliusvägen (near Slussen & Mariatorget) or Skinnarviksberget (near Zinkensdamm).
Djurgården
You can take a ferry from Slussen to Djurgården, where you'll find Gröna Lund (amusement park), Skansen (open air museum) and many other museums. This is also the place for a nice nature walk by the water, following the tram track to Waldermarsudde.
Museums and Churches
The National Arts Museum (National Museet) is close for renovation for quite some time now, so I won't recommend it until it is open again. The City Hall is pretty good (Stadshuset). I like comtemporary art too, so I'd recommend Moderna Museet and Fotografiska (make sure you spend some time taking a coffee break at the cafetaria upstairs with wonderful view of the waters). For old churches, I like the German church (Tyska Kyrka), Storkyrkan (the church where royal weddings are held) in Gamla Stan. The London St Paul Cathedral-style churches like Hedvig Eleonora Kyrka in Östermalm are also nice. I think the best way to take in the beauty of a church is by listening to a classical concert inside those churches.
Stockholm free walking tour: http://freetourstockholm.com/
This is a free 2-3h walk that relies on tips and led by local people. There are three routes: the new city (The City Tour), the old city (Gamla Stan) and the south city (Södermalm). They offer very different things, though I would recommend Gamla Stan tour followed by Södermalm.
If you have only 1-3 days in Stockholm, just travel within the city centre, there's already plenty to see. I personally don't find Drottningholm (the residence of the royal family) to be that exciting. They have a nice garden that is only nice to visit in the summertime. Other than that, the palace is quite far from the city and not much else to see when you get there.