When it comes to acquiring a bike for a motorcycle trip abroad, there are two main options:
1. BYOB (Bring Your Own Bike): As stated, this means having your own bike that you somehow get to your destination (or that is, the point at which you will start your journey). Of course if applicable you can simply ride it there, or tow it there, and finally there is the often necessary option of shipping it. For an abundance of information on shipping bikes, see one of my favorite adventure motorcycle travel forums, Horizons Unlimited, here. They have both a number of useful posts on the matter as well as a database of previously shipped motorcycles as a reference for cost.
2. BYBT(Buy Your Bike There): There is no doubt, that it is a nice form of relief to have the motorcycle all squared away for your journey before ever setting foot on foreign ground. On the same token, this peace of mind is very expensive, as shipping a motorcycle can be an enormous expenditure for your journey. Recently I chose to go the cheaper route for my planned trek through South America. Rather than riding a motorcycle down from the US, I chose to find one somewhere along the way, with 'somewhere' being a very broad statement.
Contrary to popular belief, it isn't so difficult finding bikes for sale in other countries, even if you don't speak the language. I again turn back to Horizon's Unlimited for this. They offer a forum for adventure travelers around the world to sell their rides. Check it out here.
When purchasing a motorcycle abroad, the perfect situation is to find someone from your home state, or a home state with simple or relaxed motorcycle registration policies (i.e. through the mail) to make this process easier. For me, I chose the hard route. I purchased my KLR650 from a Costa Rican national. What ensued was about two months of jumping through bureaucratic hoops to get the bike into my name. All said and done, that process cost approximately $600, to transfer ownership, purchase insurance and have the bike inspected. Other countries would be much easier. It all worked out quite well for me, but I had a strange set of circumstances (to be covered another day) and generally speaking, I would recommend anyone to avoid purchasing a motorcycle in Costa Rica in particular.
One way to get around the issue of bike registration is to get a bit creative with your documentation. Now I can't recommend this to anyone, but if someone were so inclined (and many have been in the past) you can scan the registration documentation into a computer and change the important parts. Once done, the new owner and the old owner both ride to the border, out of one country using the old registration docs (just to "check the bike out" so there is no lingering importation liability for the previous owner) and then the new owner will ride the bike in to whichever country the would like to go to, using the new documentation. From what I have heard (again not condoning this behavior) is that if you print in full color, front and back, you'll impress the border agents with legitimacy and no problems will be had. Of course, this isn't a good idea if the country which the bike is registered in is on your itinerary, because they will be more apt at spotting legit documents and forgeries.
One last option is to rent your bike, but that can cost as much as buying one of your own, save for the hassle. However certainly this is something to consider. Check back on Horizons Unlimited for some good rental businesses to consider.
One last excellent resource for adventure motorcycle travel is ADVRider.
Enjoy the ride!