The one that you already have.
Any kind of bike will do (many years of the biketour tell us so!), but clearly some bikes will make it a lot easier. In the ideal case we would recommend using bikes invented for travelling: trekking and touring bikes. If you have a mountain bike, don’t forget to put road tires on. If you have a road (racing) bike, don’t forget to put more rough tires on. This will make your life much easier, we don’t always know beforehand how the road surface is going to be.
New super-cheap bicycles from a supermarket are most likely worse than using your repaired grandparents’ bicycle. It’s a DIY tour, you don’t need to buy the most expensive equipment. Full suspension bicycles will cost you a lot more energy so think twice if you want to ride that for longer distances each day.
No matter what kind of bike you use, it’s important to note that if you want to travel without much stress. It’s best if you find a frame size that is the right size for you. It will be a long ride and you have to feel that your bike is just another part of your body. Ride it sometime before bringing it to the tour.
How many gears? We have noticed some people travelling with just one gear, but if you want to have a less painful trip (and avoid pushing your bike up hills), you should be sure to have at least some light gears. Maybe you want to take a look at the topography of the region where you are planning to join the biketour and think about whether you want to take a bicycle with racing, touring or mountainbike gears. Remember that there are always hilly parts as well!