Luxembourg City is known as a small but lovely place. I visited Luxembourg for two days and it was perfect for my first EU solo trip and potentially for yours?
If you only have 24 hours in Luxembourg I would recommend spending the day using the Luxembourg hop on and hop off bus along with the Petrusse express(train). This way you would see the new city and the old city at a great price with you as a tourist in mind. You can check out the website HERE. If you have 48 hours you may want to decide what you want to do and just use public transport though the sightseeing bus is still perfectly fine.
The main places of interest I enjoyed are:
The Case Bockmates
The Panormaic Elavator
The Central Train Station Area
Villa Vauban
Kirchberg/Utopia Mall
Place de la Constitution
Local Shop
Market 
Place de la Constituition 
Place de la Constituition
Local Bus
Villa Vauban 
Vila Vauban 
The Main Train Station 
The Petrusse Express
Case Bockmates
Down a sidestreet 
The Panoramic Elevator
As you can see Luxembourg City is very green and I loved it!
