As someone who's just struggled with getting a set of Continental Grand Prix tires on Mavic MA3 rims, there's a few things I've learned to help the whole process go smoother if you're having trouble with the instructions above:
My tube blew up or has a leak right after I put it on!
What's probably happening is the tube isn't seating in the tire properly; a small bit of tube peeking out from under the bead will either cause a puncture or explode when put under pressure -- a pinch flat. To help the tube move around in the very limited space between the tire and the rim, coat the tube in talcum powder before you put it in the wheel. The talc acts as a dry lubricant, and will help the tube seat properly.
The other thing you must do when mounting a difficult tire is check the entire circumference of the wheel for places where it will pinch flat. I start at the label on the tire and, pushing the tire away from the rim with my thumbs, look down between the bead and the rim for the tube. If you see a bit of tube, try using a tire iron and lifting the bead up, then inflating the tube a bit. Ease the iron out of the bead, and hopefully the tube will be seated properly in the tire. Deflate the tube and keep checking for pinches. When you're done with one side of the wheel, turn it around and do the other side. Be thorough. An extra 5 minutes is worth it when you've taken half an hour to get a difficult tire on.
Aargh! I can't get these damned tires on these rims!
The very first thing you should do is double check that everything is the right size. The tires could be too small for your rims. (An honest mistake; at times, it seems bike equipment manufacturers are picking sizes out of the ether instead of making and following standards.)
If it is the right size, and you can't get it on, try different positions of wheel and iron. For me, holding the wheel between my knees with the tire iron facing out and pulling on it with both hands was the only way to get my Contis in the rims.
If you really, truly, cannot get the tires in the rims, it's time to get messy. Go outside, and bring a bottle of dishwashing soap or shampoo. Coat the bead at the last section of tire you absolutely can't get in with the soap. The soap will hopefully reduce the friction on the system enough that you can muscle the tire into the rim. This is messy as hell, though, so use it as a last resort.
Albert Herring says re How to repair a flat bicycle tire: Using tyre levers to put a tyre on is just asking for trouble (pinched tubes). It is quite simple if you have any strength in your thumbs...
flamingweasel says 90% of the time you're right. Any rims wider than 3cm, or tires that take less than 120psi, you should be able to do with your thumbs. But I don't know if you've ever put 150 psi tires on 700x20C rims with no irons, but I sure as hell can't do it with my thumbs. I have no problems with my 80psi Conti Top Touring tires on 1-inch rims. These Grand Prixs on teeny little rims are totally impossible to do manually.