we've shipped bikes tons of times, in many different ways (DH used to be an amateur racer - he has 5 bikes)
We have always shipped them in cardboard bike boxes that are discards from bike shops. Find out when they have deliveries, then they'll have boxes and generally, they'll give them to you for free because they'll be going in the bin anyway. They shouldn't charge too much to pack the bike because it just involves rotating the handlebars, removing seat, wheels and pedals and padding the bike so there is no rubbing.
If you can take it on the airline as baggage, then do that. Last time I shipped a bike on the airlines it was about $80 - but that was quite a long time ago and I don't know what the deal is now.
we've always shipped our bikes in our moving shipment. We've never had any problems doing it in the cardboard bike boxes. If you have a really expensive bike and ship it frequently, then i'd suggest a good bike carrier. Those are the ones that cost $500. Otherwise a well packed bike is fine in a box. DH always puts in old blankets and stuff (once he used his pillows) to pad the bike. There is lots of room for other little stuff too once the bike is packed. It is always a surprise what we'll find in the bike boxes when we finally unpack them!