Upvote:3
The answer to about any "how do I get from here to there using public transport" question can be found on the Belgian Railway's website.
https://www.belgiantrain.be/en
The trip planner on that site can plan trips for you between any two points in the country. It understands addresses, station and bus stop names, and POI's.
So you just enter "Charleroi Airport" in the "from" field, wherever you want to go to in Belgium (or Europe) in the "to" field. Add time and date, and presto, you're set.
Such comprehensive public transport planners are actually rather common in Europe, but depending on where you come from you may indeed not be used to such planners being available, or indeed being comprehensive.
In the case of Belgium however the planner can be trusted. And it will reliably tell you how to get from Charleroir Airport to Liège.
Upvote:6
This turned out to be harder than I expected. The bus between the airport and the railway station is operated by TEC and you can find the info on their website. Unfortunately the website is only available in French and German and I couldn't find a pdf with the schedule to link to.
So here it goes to find the schedule:
The price for this bus is at the moment 6 euro.
The only real alternative would be taking the shuttle to Brussels and take the train there. It is certainly faster and maybe cheaper.