Make sure you have the right new belt. This is best accomplished by finding a part number on the old belt, and obtaining an exact equivalent. If a part number cannot be found due to a worn belt, taking the old belt in to the dealer or a parts store can help. Nothing beats a part number for cross referencing purposes, but knowledgeable salespeople can match the old belt with the new one.
Make a drawing if you are unfamiliar with the routing of the belt. While it may seem simple before it's dismantled, it will seem much more complicated once you've taken it apart.
If there are multiple belts, label them. A paint pen, Sharpie, or piece of masking tape will work well. Number them from front to back.
Have all of your tools handy. And the least, you will need to figure out or research how your belt(s) are tensioned and how to release that tension. There are generally two types of systems. #1 is the newest, the constantly-tensioned (By spring, or hydraulic means) serpentine-belt type. This generally requires you to push with a wrench or a serpentine belt tool the tensioner away from the belt, and pop the belt off any spot that is easy to reach, replace the belt, push the tensioner away from the belt again, and finishing the routing of the new belt. and #2, the component-tensioned system, which is usually found on older cars (Pre-1990s, and generally involves multiple belts, which are often v-type (They have a v-shape when looking at their cross-section, and fit into v-shaped grooves on the pulley. They are tensioned by moving either a component (Such as the alternator) or a tensioning pulley either closer or farther from the belt, thus tensioning it. You will have to loosen all locking bolts or nuts on the tensioner pulley - which "lock down" the tension you will be applying, as well as the "Pivot" bolt or nut if you are tensioning with a component such as an alternator or power steering pulley. Frequently, the v-type belts are much harder to change. Unless you're confident in your abilities and have some time to spare, consider taking the vehicle to a mechanic.
Loosen the tensioning device(s) for your belt. Sometimes, on the serpentine belt systems, there is a way to restrain the tensioner once you compress it, with a pin, or allen wrench going through a hole on the tensioner and into some spot on the engine. This you will find out from reading the shop manual or being clever.
Remove the old belt, paying attention to the routing.
Hold the new belt up against, the old belt, tensioning BOTH at the same time with your thumbs, rather like holding out a pair of paints, stretching them to their limits, to make CERTAIN they are of the same length.
Install the new belt following the diagram of the routing you drew or obtained. If the tensioner(s) are being held back by a pin or similar, then pop the belt onto all of its pulleys, and move on. If you have to compress the tensioner again, then simply put the belt onto all of the pulleys except for the tensioning one. If necessary, to keep the belt from falling off, use clothes pins, or spring loaded clamps to secure the belt to a pulley or two.
Compress the tensioner if necessary and place last un-routed bit of the belt onto its pulley, release the tensioner and let it do it's job.
If tensioned by a component or threaded adjuster, then tension the belt. A good rule of thumb is that the belt should have a maximum of 1/2 inch of movement in the middle of the belt's longest run. Check your vehicle manual for more information. If in doubt, consult a mechanic: an improperly tensioned belt can damage your vehicle.
Double check all that all belts are on their pulleys completely, are running straight (look from the side that all are in a straight line, and not bending out or in because it's going to the wrong spot on a pulley).
Make sure any locking bolts or nuts or pivot bolts or nuts that you loosened are tight.
Remove yourself and your tools from the engine compartment and leave the hood up. Turn over the vehicle and observe the belt for a few minutes to check for problems. Noise, squealing, grinding, smoking, etc. are all signs that something is wrong. The belt may be incorrectly threaded or improperly tensioned. Turn off the vehicle and allow it to cool before reexamining the belt.
If you do have problems, check that some component such as the power steering pump hasn't failed, and hasn't frozen upon you by trying to spin it with your hand. ALL components except for the crankshaft should be easily turned by hand, and spin freely. Make certain the pulleys are all aligned with one another, make sure there isn't a bunch of oil or coolant all over the belt or pulleys, and if so, fix whatever caused the leak, or if it was just you getting them wet, spray some brake clean (in a can, available from parts store) over the belts and pulleys which will remove the oil or coolant, and evaporate quickly leaving no residue).
Do not put your hand, or any part of your body, into contact with anything spinning or moving on a running vehicle.