Kathmandu is the capital of Nepal and the largest city in that country.
Hitchhiking out
At present there are only two major roads leading out of the Kathmandu Valley. One goes northwest to Tibet, while the other leads west to Mugling and Pokhara. If you wish to travel to anywhere in the Terai or cross into India, you must first go west, then turn south at Mugling.
West (Prithvi Highway)
The highway going west from Kathmandu leaves from the Kalkani neighborhood. From the intersection of this road and the Kathmandu ring road here, you can walk up the road less than a kilometer and find a place where a car can pull over.
Alternately take a bus from Ratna Park to the town of Chandragiri (चन्द्रागिरी) for 50NPR which then offers an easy choice of roads South to Hetauda, or further West towards Pokhara.
North East (Melamchi)
Take a bus from Ratna Park which will get you as far as Sankhu for 50NPR. The bus can actually take you further up to Shankharapur for around 70NPR.
From Sankhu make a sign for Shankharapur (शङ्खरापुर), and from there the road splits towards Melamchi. Make sure you don't take the Northern "Piplie" road, as when you get there you'll have a hard time getting out as there's very little traffic.