The first thing to remember when undertaking Luxembourg genealogical research is that there are "two Luxembourgs" -- the Grand Duchy of Luxembourg (an independent nation) and the Luxembourg province of Belgium, just to the west of the Grand Duchy. The "Luxembourg Culture Zone" in Europe (the area in which the Germanic dialect Luxembourgeois or Letzebuergesch is spoken) includes the Grand Duchy, part of Luxembourg Province, and small areas in the Netherlands, France, and Germany -- together about 3,000 square miles including nearly a 1,000,000 people. Luxembourg genealogical research sometimes will require searching Belgian, French, German, and Netherlands sources instead of, or in addition to, material from the Grand Duchy proper.
Reading a history of Luxembourg, particularly if you are unsure as to where your immigrant ancestor originated, often is a good introduction to genealogical research. Three standard histories are available in the department: