Nepal's busiest flight path given takes you from Kathmandu to the mountains of the Solu Khumbu, landing in Nepal's iconic Lukla airport. Flights to Lukla depart in "phases", where one phase typically consists of four flights. Your boarding pass will designate which phase you belong to and you’ll be put onto one of the flights within the phase. Kathmandu-Lukla flights — as with any mountain flight in Nepal — depend on the weather, so you'll always want to build in a day or two buffer on either end in case of flight delays.
UPDATE: Kathmandu's airport is undergoing renovation for the rest of 2019. During this time, flights between Kathmandu and Lukla will depart from Manthali's Ramchhap airport, a 4-5 hour drive from Kathmandu. Nine flights are expected to operate daily, from 6-6:30 am to 10:30 am. You can either drive to Manthali the day before your trek and spend the night (keep in mind that accommodation is basic) or you can leave Kathmandu at midnight on the same day. For those who are short on time, you can still take a helicopter from Kathmandu to Lukla.
You can also take a jeep or public bus from Kathmandu to Jiri, a town in the Lower Everest region, and start your trek from there. It often takes 3-4 days to reach Lukla, and you'll see far fewer trekkers, have most of the trail to yourself and pass through beautiful Sherpa villages along the way. Many of the guides and porters that you'll see in the Upper Everest region call this region their home.