Transporting a bicycle on a plane isn't all that tricky. If, when buying a ticket online, you can select "sports equipment," or, even better, "bicycle," you should do so, having first carefully read all the terms and conditions of the specific airline.
Usually, this pleasure costs between 30 and 40 euros one way, and they ask you to pack it (if they ask at all) in a special bike bag, a bike sleeve, or a cardboard box. Simple trash bags wrapped in duct tape will also do the trick if there are no other options, which is often the case on the way back (so you don't have to lug the packaging around with you). When packing, you need to turn the handlebars sideways, unscrew the pedals, and let some air out of the tires.
I went with the tried-and-true option of a box, scored for a "thank you" at my local bike shop, "Veloserviss," the one near the market and the central high school, right across from "Sport 2000." The box successfully fit into my dad's Volkswagen Polo, even though it's not a big car. We made it to Vilniaus Oro Uostas (Vilnius Airport) in a jiffy, and even made a detour to the Akropolis mall along the way, since my dad sincerely believes that Lithuanian bread radiates some particularly blissful emanations.