We suggest you the following itinerary: Amsterdam-The Hague-Rotterdam-Antwerp-Bruges-Brussels-Luxembourg. At least 1 day for each city, and the rest - as much as time and budget allows. Although the capital of the Netherlands is definitely worth staying at least a few days.
You can fly to Amsterdam by different airlines, the easiest way to do it is with KLM or Air France. The best way to travel between cities is by bus or train (slightly faster than buses, but more expensive). From Luxembourg you can take a bus for €5 to Charleroi, Lille or Frankfurt and fly home from there.
Amsterdam
The legendary city is the capital not only of the Netherlands, but of the entire liberal world. Millions of tourists from all parts of the world come here, fly in, come on foot. Amsterdam first of all attracts with its architecture, cultural sights, and especially with its informal, free and very "friendly" atmosphere.
If you don't have a lot of time, and you want to see the city in full, then a tour will help you: fascinating walks, sights, history of the city - all this in 3 hours.