Currently, Nepal is the only country you can leave India for by land.
If you plan to leave and re-enter through land border crossings, you'll need a multiple-entry visa. You can get a one-year visa or a 5-year visa through our visa service.
As far as I know, there is no land crossing between India and China (Tibet) open to tourists. Foreign passport holders can (in principle) cross from Amritsar into Pakistan (to Lahore)—but I wouldn't try it right now—and into Bangladesh.
There are several border crossings between India and Nepal:
The westernmost crossing is at Banbasa/Mahendranagar (on the Nepali side), about 8-10 hours east of Rishikesh. It is also the closest crossing for travelers coming from the northern Himalayas. If you're arriving from the northwest (say, Dharamshala or the Manali and Shimla area) and don't want to stop around Rishikesh, you can take a direct train from Pathankot or from Chandigarh/Ambala (if coming from Manali or central Himachal Pradesh) to Bareilly, a city about two to three hours from the border, then take a bus to the crossing. There isn't much to do in Banbasa or Mahendranagar, and Kathmandu and Pokhara are a long way away (over 15 hours of travel!). This crossing is only worth using if you want to renew your visa rather than continue traveling in Nepal.
The other crossings are better suited to travelers in central India:
The Sunauli crossing suits travelers in Varanasi and elsewhere in the central Ganges Valley, and is also the most convenient crossing for Pokhara in Nepal. It is three hours by bus from Gorakhpur, which has direct trains from Delhi and many other places. Another crossing is at Raxaul Bazaar in northern Bihar, suitable for travelers coming directly from Kolkata or Bodh Gaya and heading to Kathmandu (I used this crossing from Kathmandu to Bodh Gaya). The easternmost crossing is at Panitanki (on the Indian side)/Kakarbhitta on the Nepali side, and suits travelers in the northeast. Of all these crossings, Sunauli is the most convenient and is also relatively well organized on the Nepali side.