København, Denmark

Overview of neighborhoods

Østerbro - Wealthier neighborhood at the east and northeast of the city center. 
Nørrebro - Highly recommended streets: Jaeggersbroggade, Blågårdsgade.
Indreby - The city center is home to a number of museums; the Rådhus; Tivoli amusement park; and the main shopping street, Strøget.
Vesterbro - Old Redlight district west of the central train station, now a hip and grungy neighborhood. Home to the "meatpacking district" with restaurants, bars, and night clubs. 
Frederiksberg - Very nice and wealthy neighborhood, technically a separate city within København. 
Christianshavn - Wealthy neighborhood with small streets, canals and boats; good for a stroll.
Christania - A "free-zone" within the city. This is Copenhagen's version of Venice beach (California) filled with hippies, squatters, teens, and tourists. It also has stoner art and good vegetarian food. On a sunny day it's nice to walk around the park next to the lake or listen to outdoor concerts.  
Amager - The island with the airport, also has a nice beach (Amagerstrand), the aquarium, and some lovely cycling trails in the south.

Transit

Buses, S-train, and Metro - These are the most common means of public transit in København. The travel price is based on moving between different zones in the city. It's economical to buy an anonymous rejsekort (refillable card) from vending machines. Note: Taxis are extremely expensive.

Bike rental - The absolute best way to experience the city, and get around it's neighborhoods. I can't overemphasize the importance of a bike in making for a fun visit. Nearly all bike shops rent bikes, I recommend Baisikeli in Vesterbro. Also, city bikes are a free option. With a map and a bike you'll get the most out of the city.

Sightseeing Tours

Canal tour - Highly recommended. Gives a great over-view of the city and a unique perspective from the water. You'll here facts about the city while seeing different neighborhoods, and sites like the little mermaid, opera house, and the "black diamond" Royal Library. Find the tour boats at Nyhavn.

Museums and walks

Lousiana - Modern Art museum with a sculpture garden located north of the city. Easily accessible by train. I also recommend the restaurant for Danish-style buffet lunch and smørrebrod.

Arken - Contemporary art museum in Ishøj.

"The Lakes" - Walk around the man-made lakes, these delimit the center of the city (Indreby) and the Northern neighborhood (Nørrebro). They are very pretty, swans swim in the water and on warm days locals sit on benches along the lakes and drink with friends on the Nørrebrogade bridge. 

Rosenborg Slot Garden - The castle gardens and a lovely park.

Botanic Garden - Small but worthwhile.

Amagerstrand - The beach!

Frederiksberg Have - Another great park.

Norrebro Cemetery - Treated as a park to walk through and sunbathe in.

Tivoli Amusement park - Occasionally hosts concerts.

Dyrehavn and Bakken - A forested area outside of the city with an amusement park.

Kastellet and the little mermaid - A popular tourtist spot.

Eating and drinking

Torvehallerne - These two glass buildings next to Nørreport station house Copenhagen's fancy market where you can buy baked goods, groceries, artisanal candy, tea, fresh cheeses, meats and fish from vendors. Or stop by for a meal. Recommended Danish treats: Summerbird (chocolates), Coffee Collective (3rd wave coffee).

Brunch and cafes

Mirabelle
Kalaset
Granola
Mother
Paludan Bog og Cafe

Coffee snob cafes

Coffee Collective
Estate

Danish pastries and sweets

Meyers Bageri - famous bakery chain making incredible bread and pastries. Some serve food too.
Lagkagehuset - for pastries and Flødeboller. 
Summerbird - for chocolates, flødeboller, licorice-flavored things, one in Torvehallerne

Traditional Danish

Told & Snaps
Schønnemann
Fiskebaren

New Nordic and similar

Bror 
Manfreds and Vin - Relae's (listed below) sister restaurant, less expensive, essential new nordic. Great sourdough bread. Right across the street from Relae
Madklubben 
Nose2Tail 
Höst 
Pony 
Geist
Radio

Fancy and Super-fancy New Nordic

Relae
Kadeau
Clou
Mielke

Restaurant 108

Other tastiness

Bæst
L’appetit
- Inexpensive, yummy selection of salads, samosa, meat and vegetarian dishes changing daily. 
Aroii
District Tonkin - Delicious Banh Mi (Vietnamese sandwiches) and concept store. Recommend the Classic, B2. 
Kebabistan
Ahaaa! - Falafel, eggplant and other goodies on Blågårdsgade. 
Hachė - "Reasonably" priced burgers.
Cocks and Cows - Cocktails and burgers.
Gorms - Along with Mother and Bæst, serves the best pizza in the city.

Bars

Mikeller
Mikeller And Friends
Søernes Ølbar
Ølbaren
Lord Nelson
Blågård's Apotek (Bar/Cafe)
Charlies Bar