There´s no one definite answer for that, as it´ll depend on a long list of things. Like how much time you´ve got, and where do you plan to start, and then go where.

Sure it´s possible to ship your bike from the US to Europe and back. Across the Atlantic by sea (Eastern US to Europe West coast), I´d budget somewhere around 500-700 euros (for a mid-sized bike - and that´ll also be a factor). But remember, that the pick-up costs at the receiving end can be substantial, and they might not get mentioned at all, when you´re sending the thing. By air, I´d roughly double that figure, but then the costs at the other end tend to be lower, and processing in general is faster&smoother. And then there are some ´readymade´ packages for sending both by air and by sea, which would be easier, as the companies will handle the paperwork, but they´ll cost you more.

Then there is the possibility of renting a bike, or buying one, and possibly even selling after the trip. Once again, the selection, prices & paperwork will depend totally on where you plan to do this. In Europe both would be quite easy, but not necessarily cheap (especially renting will be expensive, and with a rental bike, you´d probably have to stay inside the EU).

I went the other way in 2009 (Western US) and in the end, decided I´m not gonna ship my own bike over, but the main reason was that my trip was only 6 weeks. Had it been around 10 weeks or more, then surely I´d have done that. If you plan to make a long overland-trip, like several months, and also many countries, then having your own bike does have certain benefits, even if shipping it is always a bit of a pain!