Since the 2015 earthquake in Nepal, the Kodari border is closed: no vehicles or foot travelers can cross the Friendship Bridge at this time (May 2018).
The Nepal-China border that re-opened in 2017 is located in Rasuwa Fort, a hundred kilometers to the west of Kodari.
There used to be hotels and restaurants in Kodari, but it turned into a ghost-town after the 2015 earthquake and the subsequent border closing. Located 4km down the road from Kodari is the village of Tatopani, which means 'Hot Water' in Nepalese. There are about a dozen guest houses in the area around the hot springs. Restaurants are plentiful as well.
Border crossing has been closed since 2015 and is still closed at time of writing (May 2018). (A new crossing point has been opened between the two countries, in Rasuwa. No re-opening is scheduled in Kodari) This section refers to the situation pre-2015.
There are two major ways of travel to Kathmandu - taxi or bus. Taxis congregate around the Nepal Customs and Immigration office and will take you to Kathmandu in 4 hours. There is normally one bus a day direct to Kathmandu but many buses to Barabise where you can change for another bus to Kathmandu. This will take about 9 hours to travel the 111km distance to Kathmandu.
Crossing into China requires you to cross the Friendship Bridge where you will a custom building on the Chinese side of the bridge. After immigration, you will find 5 km of mountain road before you reach Chinese border town of Dram. A cab ride from the border to Zhangmu usually cost ¥10 per person, with 4 or more share the cab. You can also walk and the other is to hitchhike.
You will also find baggage porters, charging an average of 40NR per porter, who will carry your baggage across the no-man's land. Be aware that is it uphill the whole way!