How to reach Kathmandu from Siliguri
There is no direct route to reach Kathmandu from Darjeeling or Gangtok. Take a bus or taxi and get down to Siliguri. From Darjeeling it takes nearly 3 hours and from Gangtok about 4 hours to get to Siliguri by taxi. Bus will take longer. From the Central Siliguri bus terminal (i.e. Tenzing Norgay Bus Terminal on Hill Cart Road) you will get bus to Panitanki (via Bagdogra, Naxalbari etc) which is the Indian side of Indo-Nepal border. It takes about 1 hour to reach Panitanki from Siliguri.
Once you visit the Indian Immigration Office here and get your passport stamped, you can either take a rickshaw or walk (for about 10 minutes), cross over a bridge and reach the Nepal border town Kakarbhitta. Here you walk up to the Nepal Immigration Office and get your Nepal visa for 15 / 30 / 90 days etc and pay accordingly (15-day visa costs USD 25, you can also pay in equivalent INR).
Note that currently the Immigration Offices at both ends operate between 6am to 6pm (timings do change). So if any of these immigration offices is closed when you reach, you need to stay overnight at Panitanki area (no good accommodations here though). So better time your visit correctly.
Once you get your Nepal visa, it's a short walk to the bus terminal. There are plenty of private operators running buses between Kakarbhitta and several places of Nepal including Kathmandu. While non-ac buses are available, I strongly recommend you take an a/c deluxe bus because of the dusty road conditions and long tiring journey.
Most buses are in bad conditions (with uncomfortable or torn out seats). The operator 'Bihani' has relatively good buses and runs a/c deluxe buses (fare 1,630 Nepali Rupees). Although they say it takes 12 hours to reach Kathmandu, it can easily take 15-16 hours or longer. They do halt twice or thrice along the way for refreshments. Buses leave in the morning and also in the afternoon. A/c buses leave hourly between 4am to 7am and again between 3pm to 5pm.
While returning from Kathmandu to Siliguri along the same route, if you find the immigration office closed, you can stay back at Kakarbhitta which has decent accommodations. Once you cross over the border, you can get buses, taxis or even shared vans to Siliguri during the day time.