I’m not sure if this is coming a little too late. But anyways, for the first part of your trip, Mussoorie is a lovely place, the surrounding areas are very beautiful and the people are friendly. Landour has some very nice views of the mountains, and there’s a viewing platform there too. For the two free weekends you have while based there, you might want to consider trips to Haridwar, a Hindu pilgrimage site. There is a chairlift to the Mansa Devi Temple, which is worth doing. There’s also a similar chairlift up to Chandi Devi, and you can buy a combination package for both at Mansa Devi. Another interesting pilgrimage site is Badrinath, in that area.
While based in Delhi, you’ll definitely want to explore Rajasthan. You could visit Jaipur, Udaipur, Jodhpur, Jaisalmer (harder to get to, but definitely worth the effort). While doing Agra, don’t forget to plan a trip to Fatehpur Sikri, Akbar’s abandoned city. Another place I would say you ‘have to go’ would be Varanasi.
There are all places in the north. If you have two weeks to also see some of south India, let me know. Goa, Kerala and maybe some parts of Tamil Nadu.
You say your goal is to not only do the touristy things. Well, for that you have to understand that India is not just a list of must see sites, but more a huge vibrant country where everyday life can be lots of drama. Its very hard to get bored in India, just observing the people going on with their daily lives on the streets, and the madness of it all, can take your breath away. Its important to observe, and not judge, make time to take in the surroundings, and maybe have a cup of chai. Talk to the locals, treat them with respect, and they will respond to you. Everything is interesting, and I would say instead of letting the culture shock you, let is wash over you.