Luxembourg City is the country's transportation hub and there are many day or 1/2 day trips via train/bus that are worth doing. My favourite was to 2 towns famous for their castles. One was Vianden & I've devoted a separate journal to it. The other is Clervaux & it was actually the easiest to get to. The train trip takes about 1 hr. followed by a 10 min. walk into the centre of town. If you have your own car, it's even easier.
Clervaux Castle reminded me of a large manor house in France's Loire Valley. It's all white with grey roof & is very pretty. There are only 3 sections open for viewing - each houses a museum. One is a war museum (closed during my visit), another highlights the Castles of Luxembourg and the 3rd, and my favourite, holds the Family of Man exhibit. This is a collection of photo portraits taken throughout the world in the 1950s. Very moving & worth the visit (Free with the Lux. card).
Behind the castle those with energy to burn can follow the road up the hill to the Benedictine Abbey of St. Maurice and St. Maur. Along the way, there are a few breaks in the trees that allow for great views of the town below. The Abbey was locked when I got there but it's a pretty sight with lots of green trees, birds chirping, etc. There is parking for vehicles at the top if you choose to drive. If you're lucky, the monks will be hosting a concert of their Gregorian chants.
Clervaux has numerous little restaurants & cafes and the town is worth at least a 1/2 day visit.