Central train station called "Hlavná stanica" is located within a few minutes from city center by public transport. The visage and surrounding of the station is awful, however, it should be modernized in a few years. Maybe you will not even believe, this is a central station in a capital city, but it is a central travelling point even for the public transport. You can get everywhere from here even during the night. However I do not suggest you to stay there during the night as it is a place full of homeless people.
Bus number 61 will take you directly to the airport in less than half an hour and you do not need to pay any special price for a ticket. This bus has also a night line N61, so you will not get stuck on the train station even between 23 and 4 o'clock.
Trolleybus number 210 connects the train station with the central bus station within 15 minutes maximum. For the list of transports departing from train station see the schedule (EN).
If you want to go to the city center, the best way is to walk down the Šancová street to the tram station "Račianske Mýto". From here you can take whatever tram NOT directing to "Rača" and you are in the city center within few minutes.
there should be direct connection from Zagreb to Vienna or Budapest. From these cities, you will get to Bratislava easily. And from Bratislava you will have to take a train to Žilina. Altogether it should take around 11 - 16 hours.
Have a nice trip.