Do you have a salad spinner? You an spin the clean cilantro dry, wrap it in a paper towel, seal it in a ziplock baggie and store in the produce drawer of your refrigerator for about 10 days.
You can also tie clean cilantro in small bunches, hang it upside down in a dark room and allow to dry for a week or so. You can store dried cilantro for up to 6 months, but I think dried cilantro loses a lot of flavor.
You can freeze cilantro by removing leaves from the stems, submerging it in water (preferably in ice cube trays) and freezing. When you'd like to use cilantro, just thaw out a cube. You can keep the cubes in a freezer storage bag for up to two months.