Most new parents get a rubber bulb syringe in their newborn kit from the hospital for this purpose, and it generally works pretty well. There are also a few new products on the market (do an online search for "nasal aspirator") that may be even more efficient at removing mucus from a stuffy little nose.
Start by squirting a little nasal saline into your child's nose to moisten and loosen up the mucus before you try to suction it out. You can buy saline at pharmacies or make it easily at home by dissolving 1/4 teaspoon of salt in 8 ounces of warm water. Make a fresh batch each day and store it in a clean, covered glass jar. (If you get your water from a well, it's a good idea to boil the water first to sterilize it.)