What kind of cycling are you doing and how suitable is your bike for it? If your bike’s spot on, I’d take it, personally – I’d hate that feeling of ‘I’d love to do this on my own bike’.
For shipping, if you box the bike up as you would for flying, it should be fine; I shipped mine out to Canada, and while DHL smashed a fair amount of stuff, the bike was hunky dory.
As for whether to ship it, it may be cheaper to just take it on your flights and store it at the airport in each case while you travel about. Shipping by ship would probably take a month or so (although I’m not certain), plus you might then have to pick it up from the port.
Just my 2p, obviously! 🙂