Great advice from Fishexpo101. In case you're looking for a down-and-dirty way of getting by with the least amount of work, here's another method:
With the car running, put the temperature on your heater to "cold". This way, you don't drain the coolant in the heater core. Shut the car off. Do not proceed until the engine is cool.
Drain your radiator. If you don't, the coolant will come out when you pull the hoses off. If the coolant is relatively new, you can catch the coolant and pour it back in. To drain, just remove the radiator cap and twist the drain plug - a tee shaped thing on the bottom of your radiator - counterclockwise, and the coolant will drain out.
You have two main hoses going to the radiator. If you still have the factory installed clamps, you just squeeze the two hose clamp ends together (vice grips work well) and pull them away from the connection.
Pull the hoses off. It may help to stick a screwdriver in there to loosen up the connection.
Slip the hose clamps on the new hoses & put the new hoses on except the upper (smaller) hose at the radiator. Fasten the hose clamps like they were before.
Fill the hose that you didn't attach until you can't get any more coolant in it. Once it is full, attach it to the radiator and clamp it. then fill the radiator and make sure the reservoir has coolant in it. If you're using new coolant, use a recommended mixture of 50/50 coolant and distilled water.
With the radiator cap off, start the car and let it warm up. When the thermostat opens up, you'll see turbulence in the radiator. Put the radiator cap on.
With the engine still running, check your temp gauge. It should start to climb. Turn your heater temp to "hot". Hot air should start coming out of it. If not, you have a bubble in your system. Turn the engine off, disconnect that hose and pour more coolant in it.
Do not start your car without putting coolant in the engine block and the radiator. If the engine doesn't have any coolant in it you can overheat your engine and crack your block.
Check your reservoir level for the next few days. top off if necessary.
If you're also replacing the heater hoses, you're better off following Fishexpo's advice.
I know that the above procedure works because my radiator sprung a leak last weekend and I had to replace it. I used the above procedure and it worked like a charm. The radiator was clean on the inside, but the outside was badly pitted and corroded. Most of the bottom fins were gone, and corrosion to the core was extensive. Not surprising, since my car is a '95 with 221,000 miles and spent most of its life in the Detroit area.