As far as being in shape/getting in shape for this trip: I should start by saying that 60 miles in a day sounds like a lot, but isn't too bad over the course of 14+ hours of sunlight. You will need to be in shape, but you don't need to be an athlete (I am not). Anybody in good health who has practiced a bit on a bike should be able to do this trip, but I will say that the more you ride now, the less pain you will be in during the trip. Frankly, the hardest thing about this trip is usually mental (example: bike all day in the cold rain and go to sleep in a damp tent)
My suggestions for "training" for this trip are:
1. Ride a bike often - the best way to get in shape for bike riding is to ride a bike (unsurprisingly). A lot of biking skills are unconscious and are learned mile by mile.
2. Remember that the goal here is distance, not speed - If you drop your speed a little when riding, you greatly extend your endurance. On long rides, I try to maintain the cardiovascular equivalent of a brisk walk. I have actually found that I have had trouble getting myself to go slow enough on long rides.
3. Practice going long distances - At least once, take a weekend day and do a 50 mile (or more if you want) ride. When you do this, remember to take your time, maintain endurance (not too fast!), and bring food. Biking 60 miles is not equivalent to doing a 20 mile ride three times.
4. Remember to eat - Make sure you eat before you go for a long ride and probably at least once during the ride - your body will thank you. It actually is a little tricky to get yourself to keep eating/drinking, but that is the mindset that you need. Most of the time, when a person is feeling tired or mopey on a long trip, it is because they need to eat.
5. Try your equipment set-up - Put all your gear on your bike as you plan on doing it on the trip and go for a ride. This will give you the feeling for how the load will affect your handling, inertia, and speed. This is a good thing to do a week before the trip: you can have everything packed and actually see how much stuff you have and how it handles.