Three languages are recognized as official in Luxembourg: French, German, and Luxembourgish (also known as Letzebuergescha, Franconian language of the Moselle region). German and French are more administrative languages. Luxembourgish is the mother tongue for most Luxembourgers.
However, over one third of Luxembourg's overall population is made up of foreigners. There are a plethora of other languages spoken, including a large number of native Italian and Portuguese speakers. In addition, there are many people that are fluent in English, but try approaching in French if possible.
There are many language schools catering to all languages and people in Luxembourg. Deterime your ability, how many hours you have to study, and how much money you can spend to find the right school. Language classes are generally:
Extensive - 4 to 10 hours per week
Intensive - 15 to 20 hours per week
Total immersion - 20 to 40+ hours per week
Luxembourg Language Centre
Established by the Ministry of Education, the Centre is affiliated with the official language training institutes including the Alliance française (French), the Goethe Institut (German) and Cambridge University's program in TEFL (teaching English as a foreign language). The center offers classes in eight languages (English, French, Dutch, German, Italian, Letzebuergesch, Portuguese and Spanish) at all levels and in a variety of formats, from full-time intensive courses to evening classes held in secondary schools.
Another option is private lessons which allows for one-on-one tutoring. This path is generally more expensive, but may a good option if you dislike the group setting or learn better individually. Expect to pay between 15 to 40 euro per hour depending on the tutor's qualifications and experience. You can advertise for a teacher in local newspapers, on bulletin boards, or on internet forums. There should be people advertising their services in these same areas.
Another option is to sign-up or search for a language exchange. People may be looking to speak your mother tongue and by writing to each other, speaking on the telephone, or meeting you can both benefit. This is best for people who are at least at an intermediate level.