Get airborne, Luxembourg
There is a very cheap bus from Luxembourg train station (if booked early € 5) via Frankfurt Hahn airport and Frankfurt Main airport to Frankfurt Hauptbahnhof, also to Charleroi Airport (and from there to Lille, Brusseles, Gent and Bruge).
If you arrive at Frankfurt Hahn or Brussel Charleroi Airports you have exellent shuttle buses to Luxembourg.
The journey from Fr. Hahn to Place de la Gare in front of the Luxembourg main station will take just less than 2 hours, the price will range from 5 to 17 Euros depending on the time of the day for your journey and how early you book your ticket.
A journey from Charleroi will take 2 hours 10 min and the price will range from 5 to 22 Euros.
The tiny country does also offer an airport which is situated half way between the capital and the German border. Flights to and from Luxembourg are from a number of European countries and a shuttle service connects the airport with the surrounding regions.
Luxembourg has its own airlines LuxAir and when you will book it before (e.g. two weeks or more) it is relatively cheap. There is a great pastries on the board :) you should try. But Luxembourg Airport is relatively small and cosy, really easy for orientation.
The cheapest way of visiting Luxembourg from the UK is not necessarily to fly directly into Luxembourg.
Often, significantly cheaper flights are available to Brussels (eg. with BMIBaby), and from there you can get a direct train to Luxembourg.
The Luxembourg trains depart from Brussels Nord Station, (which is frequently connected with Zaventem airport), and the journey takes a little under 3 hours.
Train prices are very reasonable - I paid 18 euros from Brussles to Luxembourg, and only 14 euros from Luxembourg back to Brussels.
If you fly Icelandair, there is a shuttle bus from the airport right to downtown (train station) but if you want to spend half the amount, take the local bus. Takes you to the same exact destination for half the price.
Walk, walk, walk. And when you can't walk, take the bus. Extensive, efficient and inexpensive.