For many people, stepping off a plane into Kathmandu is an exhilarating shock - the sights, sounds and smells can quickly lead to sensory overload. Whether it be buzzing around the crazy polluted traffic in a taxi, trundling down the narrow winding streets of the old town in a rickshaw, marvelling at Durbar Sq or dodging the tiger balm sellers and trekking touts in Thamel, Kathmandu can be an intoxicating, amazing and exhausting place.
There are no trains to Kathmandu, renting a car without a driver is not a reasonable option. You can hire bikes in India and ride up, but you need to be brave. Alternatively, catch a bus. They run from Varanasi and Delhi on a regular basis. Check with other travelers to find a safe bus line (Green Line & Golden Travels have been recommended), some are quite dangerous and travel at stupid speeds through mountain passes. Many hotels will pick you up from the airport if you give them advanced notice. Otherwise, you can use a pre-paid taxi (the stand is on your right as you come out of baggage reclaim/arrivals.)