A friend of mine is working as a coordinator between more than one training center, and he would like to launch a website where all these centers can offer the training courses available and people can book the ones they want if there is a slot available, while the training centers can manage/update/add the training courses they provide in the back-end. Is there a software that you recommend for that?
There's a couple things to take into consideration. Do you want to host the software yourself? If not, look into ticketing solutions like EventBrite or similar.
If you're willing to host it, there's a ton of really good solutions available as plugins for WordPress. I work for Modern Tribe, and we make one called Events Calendar Pro which you could use with EDDTickets or WooTickets to easily make this type of website very, very quickly and easily.