Shoes must fit the pedals. There are various kinds that accept cleats. They all have a very stiff sole with mounting sockets to attach the cleat to. The shoe should fit well, it should be neither too wide nor too narrow because the feet don't get much exercise while riding and can easily go numb if the shoe doesn't fit. Since good contact with the pedal is essential, they should fit snugly; choose one half size less than you would for a walking shoe.
Adjusting the cleat should be done very carefully because riding with the feet at an unnatural angle hurts the knee. Most people need to adjust the cleat so that the heel points in (of course not enough to touch the crankarms). Most people also prefer to mount the cleats just under the ball of the foot. This takes some experimentation and riding on the block to test adjustments. Don't be careless, knees take a long time to heal...
The soles of the shoes should not be flat but curved up near the toes because they can't bend during walking, being of tough carbon fiber plastic. The attachment point of the cleat must withstand quite enormous force. I have broken five out of six Shimano shoes at that point, three out of four Diadora shoes, and no Adidas shoes. Two of the Shimanos failed catastrophically, I ripped the cleat out of the sole while riding. This is dangerous. One of the Shimanos failed in another nasty way: since the top of the shoe was not very stable, twisting the foot would no longer disengage the pedal! This is even more dangerous. It seems that lately they got better, the newest model has held up for over a year. The Diadoras failed benignly, despite the broken sole they stayed ridable (but of course I replaced them anyway). My Adidases have survived much longer than any of the others and show no signs of failing, but they just recently lowered the quality (by removing padding) and increased the price, so they are no longer attractive. Today I ride cheap Performance shoes - all shoes seem to break after a while regardless of cost so I go with $40 specials.
See also an article on "Derailleurs"
