Luxembourg

I can only comment on Lux City. Here's what I did in a day:

Go to the Cathedral of the Blessed Virgin

Walk around and along the walls of the fortress in the Ville Basse (Lower City)

Take a walk through the older part of the city in the Ville Haute (Upper City) and see the Palace of the Grand Dukes, historic buildings (one dates back to the Middle Ages and was where Goethe stayed for 10 days)

See the casemates. From there you can walk down to the valley floor of the Ville Basse.

See the Museum of Modern Art, built over part of Fort Thuengen (you can take a bus from near the train station if you don't want to drive)

Not in Luxembourg but just over the border in Germany, try Trier.
Claims to be the oldest city in Germany. Dates back to Roman times. Lost of Roman stuff (UNESCO World Heritage Site), Roman Baths, amphitheatre, largest Roman building north of the Alps (Porta Negra, a city gate).
From later times, a cathedral and nice typical German-style old city centre.
Finally, Karl Marx's birthplace.

The American Military cemetery from WWII is just on the outer edge of Luxembourg City. General George S Patton is buried there. It is a beautiful setting. I actually took the train from Trier Germany to see the cemetery. Just outside the train station, I picked up a bus to the cemetery. It was a humbling experience. If you are driving, you could easily reach it.

I also found the American Cemetary quite moving. I also think Luxembourg has some of the best crossaints in Europe. The old town is fun to walk around.

The countryside north of the capital is worth to explore with lovely places like Vianden and Esch-sur-Sûre.

Luxembourg is a lovely little country. You might consider basing yourselves in a little town instead of Lux City. We stayed in Diekirch but wished we had stayed in Ettlebruck. It's a little larger and is a public transportation hub. You can leave your car parked and catch a one-hour bus or train (4 euros per person for 24 hours) into Lux City one day. Ettelbruck has a charming pedestrian plaza, good restaurants, even a good gelato shop. And market day every Friday morning.

Other worthwhile sights:

1) WWII museum in Diekirch. Highly recommended. At the end of WWII in Europe, the US & German armies abandoned tanks, trucks, uniforms, weapons, etc. in the fields of Luxembourg. Farmers gathered them up and stored them in their barns, then eventually brought them together in this museum. It's fascinating.

2) Castle in Vianden: Drive to Vianden and hike up to the castle (or you can take a chairlift part way up). Beautiful views.

3) Edward Steichen's famous Family of Man photographic exhibit, now on permanent exhibition at Clervaux Castle.

We fell in love with Luxembourg. Rolling green hills, no crowds, everything is spotlessly clean, everyone speaks English and is very helpful.

I second the items mentioned before: Vianden, Clerveaux (Family of Man exhibit), Trier, Ettelbruck, both the German and American cemeteries, and pastries and coffee at a cafe. Also, if you like white wines there are several nice vineyards and Luxembourg has several nice local beers, so stopping a at a pub or ten is not a bad plan. There is history all around Luxembourg - Maginot Line, Bastogne, the casements, ... I spent the year in school in Luxembourg back on the 70's and loved it. I've been back several times over the years to visit. If you have kids considering an excellent college experience, go to Differdange and check out Miami University's European Center. It's in an old chateau (small castle) which is lovely.