We have been to Luxembourg for 2 of the last 3 years. The east of the country has some cracking roads and scenery. We have stayed at the "Auberge Aal Vienen" in Vianden. It's an old hotel and steakhouse on the cobbled street in a town with 7 or 8 restaurants and a cable-car up to the castle. The hotel has two big garages to secure bikes overnight and is on Booking.com for about £75 per night for a double.
Just south of Vianden, following the river, is Echternach, a bigger town on the German border. There's a 12th century monastery, (founded by a Geordie), even more shops, restaurants and hotels, and a town square that always hosts car shows, bike shows or beer festivals at the week-ends.
Luxembourg city is split into two by a great ravine, from memory all the big shops and squares are on the north side and the older town is on the south side.. Personally, I'd stick to the pretty villages. hills and forests to the east and leave the big shops to the women-folk. (No offence intended)
Remember that petrol is very cheap in Luxembourg too, something to do with tax concessions. Regular visitors will tell you that you know when you've crossed into Luxembourg, cos the 1st thing you'll see is a massive great petrol station with 36 pumps in use !