The following writeup is entirely subjective. All the usual caveats apply.
When visiting the capital city of Denmark, Copenhagen, you may have several different objects in mind. You could be a run-of-the-mill camera-wielding tourist; an epicure seeking out the best in local cuisine; a history enthusiast, eager to learn; or even just be looking for a good shopping experience.
The following writeup is my humble attempt to provide you with some suggestions for your visit. I'll always be happy to answer questions, and to modify the writeup accordingly.
All directions are given with the Hovedbanegård (Copenhagen Central Station) as the point of origin, except where specifically noted.
- Café Sorgenfri
Nice little restaurant in Brolæggerstræde, that serves Danish style smørrebrød (what the rest of the world calls smorgasbord). Highly recommended, but it may be impossible to get a table during tourist season.
Get there by: Following Strøget (see below) to Brolæggerstræde, 15-25 minutes.
- Botanisk Have (The Botanical Gardens)
Located near Nørreport Station, the Botanical Gardens are one of Europe's finest collections of botanical specimens. Well-designed and with many comfortable nooks and crannies to rest in, they are definitely worth seeing.
Get there by: Train, S-Train or bus (1) to Nørreport and walk, 2 minutes.
- Det kongelige danske Haveselskab (The Royal Danish Horticultural Society)
Located in Frederiksberg, an inurb of Copenhagen, the gardens of the Royal Danish Horticultural Society make up an oasis of meditative peace. Warning: I am reliably informed by people who can get lost (I cannot, myself) that it is easy to do so in the gardens...
Get there by: Bus (1) to Frederiksberg Rådhus and walk, 5 minutes.
- Zoologisk Have (The Copenhagen Zoo)
Also located in Frederiksberg, the Copenhagen Zoo is well-stocked. Recent innovations include a tropical house (dig the swarming butterflies) and a spanking-new giraffe house that gives visitors the sense of being up close and personal with the giraffes ("My neck may be longer than you are tall, but hey, mi casa es su casa").
Get there by: Bus (6) to Frederiksberg Have and walk, 5 minutes, uphill. Or, go to the Royal Danish Horticultural Society and walk from there, 5 minutes, uphill.
Okay, so I'm a tourist
- Den lille Havfrue (The Little Mermaid)
Perennially popular with the tourists, the statue of the Little Mermaid is located in the harbour of Copenhagen. Be warned: many tourists are disappointed to find that the statue is not larger-than-lifesized.
Get there by: Bus (1,6) to Esplanaden and walk, 5 minutes.
Definitely the most interesting of the traditional attractions of Copenhagen, the Tivoli gardens are an amusement park (with heavy emphasis on the "park" bit) located right next to the Central Station. Any visit to Copenhagen (touristy or not) ought to include a visit to Tivoli.
Get there by: Walk, 5 minutes.
Another amusement park (this one with more of a "Coney Island" feel to it) located some distance north of Copenhagen. Well worth the visit.
Get there by: S-Train to Klampenborg and walk, 20 minutes throuh beautiful woodlands.
Buy! Buy! Buy!
(Note: If you're from outside the EU, check your options for duty-free refunds before making any purchases)
Strøget is justly famous as mile-long pedestrian street (the model for all later copies), filled with all sorts of shops. It runs from Rådhuspladsen to Kongens Nytorv, with the shops getting more expensive as one approaches the latter.
Get there by: Walking to Rådhuspladsen and starting from there, 5 minutes. The walk to Kongens Nytorv along Strøget takes up to two hours.
- Magasin du Nord
One of the two largest department stores in Copenhagen, Magasin caters to the medium-income shopper.
Get there by: Walking along Strøget to Kongens Nytorv.
The other large department store, Illums caters to the more well-heeled shopper (they specialise in customers who have no kids to drain their cash).
Get there by: Walking along Strøget (located about 3/4 of the way to Kongens Nytorv
While you're visiting the Little Mermaid, you might want to drop by the surplus/outlet stores that have sprung up in the old warehouses along the harbour. Potentially good deals to be made.
Get there by: Visit the Little Mermaid and walk from there, 5 minutes.
(More to come as I think of it)