I want to visit Denmark (specifically Copenhagen), but I would like some advice on things I should see while I am there. What are the "must see" attractions in the Copenhagen area? Would it be best just to go on a tour that visits these areas, or is Copenhagen reasonably easy for a foreigner to navigate?
Not to worry, a lot of people there speak English, and it's very easy to navigate, with trains to most out-side-of-center featured attractions, and those in the center are all easily walkable.
http://wikitravel.org/en/Copenhagen has a list of the best things to see and do. From personal experience after two trips there, Tivoli themepark is well worth a visit. It's more old-school and not as tourist-trappy as Disney or Universal, and is right next to the Central station, so easy to get to.
A water boat tour is well worth a trip, you get to see a lot of the city from one of those, and it's very relaxing.
The mermaid is worth the photo op, if she's there - she was away on display last year.
The palace is the royal residence and looks amazing. As are many of the buildings in the center - the architecture is a boon for those keen on stuff like that. For me it was just great photo opportunities :)
For fun you can take a train across the ocean(!) to Malmo, Sweden. It doens't take long, and the ride is quite an experience.
If you like shopping, Strøget is the longest pedestrian street in all of Europe, at 1.1km, and is the top shopping area. There are also a variety of flea markets and stalls all over the place, especially in the docklands.
Also if you're a foodie, there are some fantastic restaurants, whether on a budget or willing to splurge on the top places.