Domestic Travel
We recommend booking travel agencies via TripAdvisor in tour based formats for famous sites like the Taj Mahal that avoid public transport and offer you a private driver. For reference, a typical Taj Mahal day trip from Delhi might cost around $50-60 USD pp.
Typically Western 5 star hotels (Think Hyatt, Marriott, etc) offer day trip tours within the city they are in in very large cities like Delhi and Mumbai. These will show you local shopping spots and some historic sites. For example, we’ve heard good things about the Delhi day tour from Radisson Blu Plaza Delhi Airport Hotel. Check with hotels you are booking to see what they might offer.
Within a few hour driving distance or 1 hour flying distance from the wedding, Mumbai or Goa are good tourist destinations.
Agra, Jaipur, & various cities in Kerala are good historical sight seeing options within a few hour flying distance from Pune.
The Himalayas are a lot farther and their peak season is more like April/May. They are massive, so a lot of the very grand sites might require hours of driving and/or hiking to see. There are some accessible locations in November though, like Shimla, Manali, and Rishikesh.
For the beach, look into Goa, Lakshwadeep, or the Maldives.
You’ll be in the third largest mountain range in India (the Sahyadri range) for the wedding. Since you’ll get a dose of nature and the outdoors while you are there, other nature heavy sites might be less worthwhile to prioritize.