I'm flying into Copenhagen and will spend a few days there, but I've heard the trains are pretty good and want to venture out to see some other parts of the country, so any suggestion would be much appreciated! Any Copenhagen suggestions would be awesome, too. Thanks!
First Copenhagen: Plan on drinking a lot of beer and coffee since Danes drinks more per capita than anywhere, i was told. You should rent a bike (i got one for less than $10/day) for at least a day and ride around the city like a local. Overall the country is pricey, but very worth it. It's clean, beautiful, historical and friendly.
We stayed in this apartment we found on Airbnb in the neighborhood called Nyhavn. It's the quintessential Copenhagen canal with lots of resturants and cafes to sit outside and eat/drink. I HIGHLY recommend, MS Donna Wood It's the old ship near the bridge. You can just go on board, grab a couple beers down below, and then go up to the roof, set up your own table and watch the Canal Tours go by. At the end, just take your beer caps downstairs to pay for the beers you drank. It's only 20kr per beer as opposed to the 40kr in one of the cafes across the street. I hit this place twice :)
Early every morning we took a jetlag induced walk across the canal to the neighborhood Christianshavn because we found the pastries at Lagkagehuset to be the best in the city... and I tried a lot. We'd also get a roll with some havarti cheese to, ya know... be healthy. Grab one of a few bar stools along the window to look over the canal and watch the city wake up and bike to work. Freetown Christiania (the tax free co-op living area most noteworthy for is ability to buy/sell drugs (mostly pot) on what is known as "pusher street") is also nearby and worth a stroll through. In the morning it's very quiet and non-druggy at that early hour.
The Canal Tours Copenhagen, which leaves from Nyhavn, came highly recommended and it was a nice way to see the city, although i was hoping for more information from the tour guide. Also, I wouldn't recommend the hop on/off version of the canal tour as it didn't look like a lot boats came around because at each stop they were waving at us to come pick them up.
I'd say the view from Rundetårn (Round Tower) is worth the walk up (800ft tall with no stairs, but a circular walk to the top) and it can combined with a stroll around the Rosenborg Slot (Rosenborg Castle) as they are nearby. Also, at the bottom of the Round Tower is the organic hotdog truck called døp which is one of the best polser/hotdogs in the city. I had a goat saugage which was off the charts. From there you can walk back to Nyhavn and explore the walking streets and many squares.
I'm sure you'll go check out The Little Mermaid, fairly underwhelming IMO and Tivoli which was fun mostly just for kids so I'd recommend doing it at night if there is a good concert playing. Sting was playing there around the same time we were visiting.
Noma has been the #1 restaurant in the world for a few years now, but requires a reservation months in adance and might require a second mortgage. It was in the middle of rennovation, but i did get a private tour from the head sommelier, which was sick. Also, Noma former head chef, Matt Orlando, is opening a brand new place called Amass which will be on every foodie's list as well. For an open-face sandwhich called smorrebrod (a Danish favoite), many people will recommend Restaurant Ida Davidsen v/A Siesbye, but my foodie friends in the city swear the best is Restaurant Schønnemann. It was great! Make sure you get a "reasonable" order of snapps to go with your sandwich, trust me.
Getting out of Copenhagen: There is a train station very close to the center of Gilleleje, but we rented a car and drove up. Once you're up there there is a lot do and see. In the town there are some cool little harbor side restaurants.
For traditional Danish grub check out Gilleleje Havn and step into a plate of ribs. Full rack only. Don't bother asking for a half rack like I did. Sit outside and have a beer next to the boats and have fish and chips at Adamsen's Fisk or a burger and hotdog/polser at Speedy's I found the best bakery in Gilleleje to be ÅBageren, i was having a $50/day pastry habit on this trip.
You can head over to a couple near by towns worth a visit. Rågeleje City is a cute little town just west of Gilleleje and a little further is Tisvildeleje which has to have one of the nicest beaces in Denmark. I was actually blown away at how big it was and how nice/soft/fine the sand was.
We rode bikes from Gilleleje down to Hillerod to see the Frederiksborg Slot. The ride was about 25 miles through the farmland and forrest. It was a pleasant ride and extremely scenic. I would highly recommend it, however you really need a local to help you navigate the route... i wouldn't do it alone. The castle in Hillerod is thought to be the most beautiful in the country and after riding all that way to get there, i was almost magical finally seeing it. This is something I would do again for sure.
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In Copenhagen, you should visit the Nationalmuseet. Tons of local history and Viking artifacts, a pretty fascinating museum.