There are 2 airports close to Liege: Brussels Airport and Brussels South-Charleroi. Bruxelles South is located in Charleroi, 80km South of Brussels. Both airports are located 100 km from Liege.
From Brussels Airport
The main airport, Brussels Airport (also called Zaventem Airport) links more than 200 cities. There is a good chance you land here. To reach Liege, you have 2 main options.
You can board a train at the airport train station, located in the underground floor. There is a train for Liege every hour. Note that you might see "Luik" (the flamish name for Liege) instead of "Liege" sometimes. The trip lasts 60 minutes and costs 20€. You need to change train in Leuven. The switch is fairly simple.
Once you arrive at Liege-Guillemins, train station, you can take a taxi or a bus to your housing.
Taxis are of course available at the airport. A trip to Liege will cost between 150€ and 200€.
From Charleroi / Brussels South
Charleroi airport is smaller, and home to a few low-cost airlines.
There is no train from the airport, but there are direct buses to the Charleroi South train station. The bus transfer takes 20 minutes. The train from Charleroi to Liege lasts 60 minutes. The overall ticket costs 20€, which includes the train and bus transfer.
Once you arrive at Liege-Guillemins train station, you can take a taxi or a bus to your housing.
Taxis are available at the airport. A trip to Liege will cost between 150€ and 200€.