Find a guide service. This is easy in Kathmandu--go to the Thamel Market area and simply walk the streets. Signs for guide services abound. Duck inside the various shops and offices and get details and prices until you've found the one that you like. Any guide service located in Thamel is reputable and has likely led thousands of tourists into the Himalayas before. The guide service is recommended because it will take care of everything for you--accommodations during your trek, all of your meals, various entrance fees, and your guides, of course. The entire trip (from Kathmandu) should cost you between $250 and $500.
Fly to Lukla. This is your starting point. Spend the night here to begin your acclimatization.
Trek from Lukla 3 hours to Phakding. Spend the night here, then trek another 5 hours to the picturesque Namche Bazaar, a village that clings to the terraced mountainside at about 11,200 feet. Spend two nights in Namche Bazaar in order to acclimatize. There's lots to do here, including several nice day hikes, visits to tea houses and even a small local museum.
Trek 5 hours from Namche Bazaar to Tangboche (12,687 feet), home to an impressive monastery, where you'll spend the night. Wake up early and trek another 5 hours to Pangboche (14,000 feet). You'll spend two nights here as you continue your acclimatization. The village offers several day hikes.
Trek another 5 to 6 hours to the village of Lobuche, at a stunning 16,200 feet, where you'll spend the night before the big push to base camp. The next morning, trek 7 hours to Mount Everest Base Camp. Here you'll wind through the mighty Khumbu Glacier and get as close to the tallest mountain in the world as possible without actually climbing it.