There's no train connection with Kathmandu Airport.
Car and bus
The airport is situated 6 kilometers east of Kathmandu's city center. Since the airport is located on the city's Ring Road, it's easily accessible by rental car. From the city center Dilli Bazaar Road which turns into the Sinamangal Road leads to the airport.
On this Ring Road there's also a bus stop where several buses stop. The bus to the center costs approximately 40 rupees.
Car rental Kathmandu Airport
You can find counters of car rental companies in the Arrivals Hall. Several rental companies have a presence at Kathmandu Airport, including international providers like Avis, Budget and Hertz.
A taxi from the airport to Kathmandu's center costs 250-300 Nepalese rupees. The ride takes twenty to thirty minutes. If you walk from the terminal to the road (Ring Road) you can often stop a cheaper taxi. Plenty of people are willing to carry your baggage for a fee.
There's a short-term parking lot outside the terminal building. Seen from the entrance, the long-term parking lot is located further to the left.