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.
I second the upside down drying out method. We haven't done them in a dark room so I don't know if that is a requirement or not. We usually rinse them, dry them and tie them upside down in the garage for a few days.
you can improvise a dehydrator using a box fan, furnace or air conditioner filters (foam square ones) and a bungee cord. search for it on you tube. Alton Brown from the food network was where I learned that trick.
Also you could turn it into Cilantro pesto by pureeing it with Olive oil. it will keep for several weeks that way when put into a sealed jar.
Freezing maybe, but still some flavor is lost. Drying, I do not recommend. Even if you're careful about it, I'm telling you it loses all of its flavor. I didn't even know about dried cilantro until a couple months ago, and there's a reason: it just doesn't work like other herbs.