The Roman Catholic Cathedral is in the centre of Luxembourg City and one of the main stand out attractions. This is in fact the only cathedral in the entire country, and displays great architecture both inside and out.
The second city which can easily be reached in 45 minutes via the many buses that leave the capital.
Place Du Marche is the main central square and a great starting point to explore the city. The Abbey of Echternach, City Hall, and Column of Justice are some of the main highlights, but don't expect too much. Walk the cobbled streets, drink coffee, and eat in the local bakery.
The MNHA is by far the top museum in the country. displaying artworks and artefacts from all epochs of Luxembourg history. There's also plenty of unique objects and displays within the multiple rooms. The exterior of the building is impressive, so be sure to take a 360 degree walk around the building to view the rear.
Okay so it's a lift, but it's a very cool one. Great engineering design, and it's a free attraction which offers you great views of both lower Pfaffenthal and upper areas of the city. The hardest part is perhaps locating the lift. Use "Fondation Pescatore Retirement Home" as a GPS location. You'll find the lift within this park. There's a glass floor, which is pretty cool too!
Due to Luxembourg becoming a popular financial hub, you'll come across lots of grand buildings. These are mainly Investment Banks and Financial Regulators, but if you're an architecture geek, some of them are very impressive!
The casements of the city are impressive, especially as the passages span across 21km. This is the central point of Old Luxembourg, and has been declared a UNESCO site following the amount attacks the city has survived over the decades.
You're looking at a €6 entry fee to go inside the casements themselves, otherwise just admire the flower displays, bridge views, greenery, and even a skater park.
Luxembourg is actually a city where it's worth contemplating doing the very touristic "Red Bus Tour". Due to the size and shape of the city, everything is pretty spread out. Yes, you can walk it, but the bus stops at all of the major attractions, has a good audio guide, and is a cheaper ticket cost compared to other European cities.
Accommodation isn't the cheapest in Luxembourg either. The best I could find was Grand Hotel Cravat, which is based on my location:cost ratio that I generally use. The hotel is directly in the centre of the city within easy walking distance of both the old town and casements. It's certainly no 5-star place either. There's an outdated, yet old skool elegant feel about the place.