On the Bicycle Touring List, there was a discussion of various bicycle repair manuals and resources. Mark Boyd, who has toured many thousands of miles, offered this perspective, which I've reprinted here with his permission.
I read books about bike maintenance back around 1970 when I first got into bikes and learned a lot from them, but I already had a strong background in doing fairly major - work that I wouldn't dream of attempting today ;-} - repairs on automobiles. Bicycles are much simpler and easier to work on than cars, so it was easy for me to move from repairing cars to repairing/rebuilding bikes.
The fact that I already had a pretty good set of tools helped, but the main thing I had was mechanical empathy. I already knew how to relate to a wide variety of mechanical things. You can't get that from books. You get it from building and working on things, if you get it at all, and most people, including most bicyclists, never seem to get it.
As others have pointed out in this thread, you only need a small set of skills to handle the large majority of things that happen on a tour. The three most common issues I have had to deal with on tour are:
1. Fixing a flat tire. You need to carry tire irons (2), a patch kit, a spare tube, and a pump. I've been know to carry multiple sets of all of these on tours in remote places.
See: http://sheldonbrown.com/flats.html for a good tutorial on fixing flat tires.
2. Replacing a derailleur or brake cable. The difficulty in doing this is dependent on whether it a front or rear cable and on the type of shifters you have. The only tool required is the appropriate - carry a set - allen wrench(s) if you have cables pre-cut - by you or your bike shop - to the right length.
After you have physically replaced the cable, you also need to adjust the derailleur or brakes. This can be a real pain, especially with indexed shifters. This repair definitely needs to have been practiced before you need to do it on the road. If you know what you are doing and have precut cables, it takes maybe ten minutes. If you have never done it before, you'll need incredibly good mechanical empathy to be able to figure it out without someone, who has done it, helping you. Get that help before your tour and carry pre-cut cables!
3. Replacing a broken spoke. Now we are getting into a area that really require some skill, although you can usually - I have, several times - kludge it well enough to get to the next bike shop. Replacing a spoke may require a tool to remove the rear cogs as well as a spoke wrench and tire tools. All you need to do is remove the wheel, remove the broken spoke, put a new spoke in it is place, and true the wheel. You should carry three sizes of spokes, one for the front - I've never broken a front spoke - and one for each side in the rear. The ones that are most likely to break are the rear, drive side, spokes. They are also the ones that are hardest to repair.
Removing a front wheel is easy - loosen the brake, open the quick release and loosen it - damn lawyer lips! - and take the wheel out. The back wheel is more of a pain because of the chain and the derailleur. It is a good idea to get somebody to show you how to remove and replace the back wheel. If the broken spoke is on the cog side of the back wheel, you'll also have to remove the cogs. This takes a special tool called a hypercracker.
Truing a wheel takes skill that, at least for me, only came after I built my own wheels. Before that, I was able to get the wheel functional and ride on with a wobbly wheel (and lousy brakes) till I found a shop to true my wheel.
These three skill sets would take care of 99% of the mechanical problems I've had on tour. Of course, it is a good idea to minimize the chance of having these problems even if you are prepared to deal with them.
The broken spoke problem can be minimized by using good touring rims and spokes and proper wheel building technique. I haven't broken a spoke it my last 30000+ miles of touring.
The broken cable problem can be minimized by replacing the cables before the start of each tour. I don't bother since I can fix cables easily on the road. Of course there was the time I broke a front brake cable on tour...
The flat tire problem can be minimized by using good (Shwalbe or Conti) touring tires and by using flat resistant tubes and/or tube sealants. It is also important that you pay attention to the road ahead and learn to avoid, rather than ride through, debris on the road. I average > 10000 miles between flats on tour.
Most tourists do not seem well prepared for dealing with these common problems. I think the best way to be prepared is to have worked through each of these things with someone who does know how to deal with them. If you know someone who knows how, get them to teach you. If not, try to find a bike shop that has classes on bike maintenance and repair and take as many classes as you can.
Get a book too, and study it for background knowledge before you actually work on each of these repairs. And articles on Sheldon Brown's web site at http://www.sheldonbrown.com/repair/index.html have most of the background information you'll need.