Bring a reclining lawn chair to a dark site with an open view of the sky. No trees or buildings should intrude into your view except maybe at the very edges. Depending on the time of the year you may want to bring a sleeping bag for protection against cold, dew, and mosquitoes. You'll also need a watch and a dim, red-filtered flashlight to read it by. You can make notes with a clipboard and pencil, but much better is a tape recorder with a microphone switch. This way you can dictate notes in the dark without taking your eyes off the sky.
Give your eyes at least 15 minutes to adjust to the dark. Settle in, look up, and relax. When you're ready to begin watching steadily, note the time to the nearest minute.
The simplest project is just to count the number of "shower" (S) and "non-shower" (NS) meteors that you see. Shower meteors will seem to come from the radiant of the particular shower you are observing. The name of the shower will tell you the general location of the radiant. For example, the Perseid meteor shower's radiant is in the constellation Perseus.