
Hamburg is the second largest city in Germany, after Berlin, and one of the major industrial cities in Europe. It's usually overshadowed by more touristy cities in both Germany and Europe, such as Munich or Amsterdam, even though there are a few sights worth checking out in Hamburg too.
I was personally not a huge fan of Hamburg as a tourist city, but the location of the city makes it a prime base for exploring Northern Germany. There are many day trips available from Hamburg, all of them reachable by train. Hamburg is a major train hub and you can reach most parts of Central, Western, and Northern Europe by train from the large Main Station, in German known as Hauptbahnhof, which in Hamburg is unusually central in the City.
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The number of days you need to explore Hamburg is, in my opinion, quite few. When I went there I felt done after one day. You can do Hamburg in one day without stressing through. You might need an additional day if you are interested in entering the Miniatur Wunderland, one of the major attractions. It's mostly aimed at children though. Two days is probably the ideal, as you would get extra time to just relax and take it very slow. Anything more than that and at least I would have been feeling restless.
The good thing is these great day trip options from the city. By going on day trips, you can extend the time in Hamburg to up to a week. I have visited all three of them, but only one as an actual day trip. The other two were on separate trips but still well within the train network of Hamburg and reachable in approximately one hour of travel.

Lübeck
Lübeck
The former Center of Trade and Capital of the Hanseatic League which saw it become one of the most important cities in Europe for several centuries is one of my favourite cities. Unfortunately, much of the history was erased during World War II but some important monuments still exist, such as the famous Holstentor. The city gate from 1464 is one of only two remaining city gates in Lübeck.
The Old Town, located on an island in the Trave River, has been a UNESCO World Heritage Site since 1987, making it one of the first such sites in Germany. I find the Lübeck Old Town to be very cozy and this is actually a city I've visited many times, partly due to its convenient location for a stop when road-tripping from my hometown of Stockholm. But there's also a reason why Lübeck is always chosen as that stop when there are many other cities and towns within a short distance, Hamburg being one of them.
To get to Lübeck from Hamburg by train takes around 50 minutes and has you traveling a distance of around 60 km (37 mi). The Train Station in Lübeck is located less than five minutes from the Holstenstor and the Old Town.
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Bremen
Bremen
Just like Hamburg, Bremen is a city-state within Germany. The smallest state both by area and population. It's the second-largest city in Northern Germany, after Hamburg. If you, like me, prefer city trips and to explore urban locations, Bremen is the best option from Hamburg. Visiting Bremen as a day trip from Hamburg is doable. The important sights are all within a small area and the city is very walkable. But if your schedule has room for it, staying for one night could be well spent in this city.
The first thing you should do is to head to Marktplatz (Market Square), where you can find several of the main sights surrounding the square. The famous Roland Statue and the Brick Gothic/Weser Renaissance Town Hall (both UNESCO World Heritage Sites) are both located there, as well as the Bremen Cathedral and the Town Musicians of Bremen Statue. Just a short walk from the Market Square is the neighborhood of Schnoor, which is the medieval center of Bremen, which survived World War II almost intact, something that sadly was not the case for most parts of the city.
It takes about 55 minutes to go from Hamburg to Bremen by train with ICE high-speed trains for a cost of €25-30. You can also travel on regional trains which will take about 70 minutes, but from what I've seen, the price for this is pretty much the same as the high-speed trains. The Main Station, Hauptbahnhof, in Bremen, is located a short walk from the tourist areas. The area around the main station there is not the nicest so there's no reason to hang around there, walk southwest towards the city center.

Lüneburg
Lüneburg
Lüneburg is much smaller than the other two, and it's the city that I visited on a day trip during my Hamburg stay. With a population of just 77.000 people, it would be just another suburb to Hamburg if it were not for that it is one of the larger historic towns in Germany that survived World War II intact. The Old Town of Lüneburg is authentic, which is rarely the case elsewhere in Germany.
There's a lot of beautiful architecture to adore along the Main Street of Am Sande. You can find great views of the area if you go up into the Water Tower Observation Deck (Wasserturm), and it's not a boring modern water tower, it's a red-brick cultural heritage tower from the early 20th century. Stintmarkt and Marktplatz are also important points of interest in the town. Lüneburg is however the only one of these three where I wouldn't consider taking a night. The town only takes a few hours to go through and you will be back in Hamburg before the evening.
To Lüneburg, you can get in 30-40 minutes by train. The Train Station in Lüneburg is located about ten minutes from the Old Town. The station is rather small and is essentially a small commuter station within the commuter rail network of Hamburg.
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