Traveling to India for the first time. Need help.?

From New York, I will be traveling alone to the following cities: -Delhi -Jaipur -Agra -Varanasi -Mumbai -Himalayas Questions: (1) Can I do this itinerary without a guided tour, or is this just too much to do alone? (2) How do I travel between cities in India? By train? By plane? By bus? When I visit multiple cities in Europe, I book train tickets online with Eurorail, and it's simple. Is it different in India? (3) If visiting all these cities by myself for the first time would be logistically overwhelming, can you recommend a group tour that is inexpensive? (4) Where is the best place to visit in the Himalayas? I'd like to get a good view of these mountains, and perhaps do some hiking for a couple of days, but nothing too serious. (5) Should I fly from New York to Delhi, or New York to Mumbai? (6) Finally, when are the best months of the year to visit the places I mentioned? FYI: I will be in India for about 2 weeks.