Stockholm

Some more information about what you enjoy would help narrow down the range of options. I highly recommend the City Hall, Vasa Museum, Royal Palace, Sergels Torg, and, since you have a few days, an "utflykt" to the Archipelago. Finnhamn is very nice and rustic and you can rent a cabin out there on the island for the night (as well as a sauna). If you have kids, or are interested in Swedish heritage, Skansen is a nice open air museum with a good collection of typical traditional Swedish buildings and handicraft from all over the country. Further out on the same tram line that takes you from the central city to Skansen (and the Vasa Museum) is Waldemarsudde, which is an old royal estate and art museum. The photographic museum is newer and highly regarded.

Food is tricky because it is rather expensive to go out. A typical main course will be between 120 and 200kr at a normal place. That's $16 to $27 a head (tipping is not required). A beer or glass of wine is usually about 55-90kr (unless you go to divey places or order "Falcon", which is like Bud Light). That's $6 to $12 a drink. When you live there this starts feeling "normal". Lunch is usually about half that (70-120kr) and is the best deal for eating out. Traditional Swedish cuisine is called "Husmanskost", which means "House-man's food", and is stuff like meatballs, liver, fish, and the like. Restaurant Pelikan is highly regarded for this and very nice inside. Today, Sweden is much more multicultural and what is "Swedish" food is quite different. Kebab, pizza, and hamburgers are part of the national diet. Amida kolgrill is popular and good kebab place on Södermalm. Aki Sushi on Södermalm is the best sushi in Stockholm (Råkultur is not up to NW standards). Pizza is on the lower end and so long as you get pizza outside of Gamla Stan you should be OK (but there is better types of food out there). Avoid Mexican and Chinese.

I highly recommend you look at Rick Steves Scandinavia (or just the Stockholm Snapshot Guide, if you're not going to other places in the region). I loved the Millesgarden, liked Drottningholm, enjoyed the National Museum, and found the Vasa very interesting. But most of all, I just liked walking around Stockholm - it's just a very pleasant place to hang out and explore. I wasn't as taken with the museums connected with the Royal Palace, but many love the armor and the royal coaches. As for "best restaurants," what do you mean by that? Cheapest, fanciest, highest quality food, nicest ambiance, most convenient to your sightseeing, best value? As noted above, while lunch can be a very good value due to ubiquitous lunch specials, dinner can involve sticker shock if you're not prepared.

The other tour ideas are good, make sure to include just strolling Gamla Stan. Not sure if a tour of the Royal Palace was included. There is also the daily firing of the cannon across from Gamla Stan and the Royal Horse Guard parade. For food, I defiitely recommend lunch at the market hall, I forget the name but it's in all the guidebooks (which you should have!). Pretty economical. I though RS had some economical but nice dinner restaurants in his guidebook.

Vasa was a highlight for us. Skansen was a disappointment, primarily because the employees knew next to nothing of Swedish history/traditions, they were in traditional clothes and that was about it. There were also few entertainment options during our day. In four days, I might plan to see it. In less, I'd skip it. Gamla Stan was worth a look around. There's another spot I wished I spent time at, maybe part of Djurgarden. It was essentially a giant park that seemed like a decent place to relax on a nice day (I think originally a palace and garden). The archipeligo is worth seeing, it makes sense as part of a ferry to Helsinki or Tallinn. It's about a four hour ride from Stockholm (maybe more) to get to open ocean.

Douglas: For food, I defiitely recommend lunch at the market hall, I forget the name but it's in all the guidebooks (which you should have!). There's a couple. Östermalmstorget has the most character and charm, plus the food there is very good. That's most likely the place that is in Rick's book. There's another hall by Högtorget, but it has a different character. It's more blue collar and lacks the charm of the old brick building. However, it is more interesting in terms of the mix of cultures: Högtorgshallen has more non-Swedish foods, including sushi and kebab. Izmir Kebab is a good and cheap place in there (but it is not high dining. Not by a long shot!).

The market hall is Ostermalms Saluhall, and it is a wonderful place to purchase good Swedish food or to have a sit down meal. One of the things I would do is visit the VASA museum, where the restored warship VASA is located.

I agree with the others that Vasa is great, and we enjoyed Skansen with our little kids. The harbor itself is beautiful, consider a boat trip out to the archipelago. My favorite neighborhood is Sodermalm, great place to walk around, good local shops, bars, restaurants. Mariatorget Square is a fun place to relax.