Actually, it looks like you already bought a "package" for New Delhi, Agra, Jaipur, Jodhpur, Udaipur, and Mumbai. with a side trip likely for Varanasi.
That is ok for a first time to India, really the first time is more about getting a feel of the logistics, overcoming "culture shock" in some cases (this will NOT be like the experience of your local temple, and there are all sorts of Indians not just Hindus).
So no, this does not need to be a big religious moment like specifically going to Varanasi or Haridwar with vast temples, tirthas etc.. But even with this "grand tour package" there will be no avoidance of great religious moments.
My wife can give great recommendations regarding Mumbai, I can chime in on the others but you will probably often be leaving from your booked hotels with others (both foreigners but you will be surprised at the large number of locals and Indians on your booked bus tours) so you will be going to fixed locations.
But the Indian state run tours out of hotels (charted with private tour agencies) are actually very good compared to the USA actually, they get you to the important, exciting and often very religious sites. When I get a chance, I might suggest some sites in the locations you list, but the tours for a first timer are not bad.
There are lots of little tour agencies and transport agencies run from small shops near every hotel you stay, these are quite good actually but pay only cash rupees.
Give me a chance I will post more, but IN GENERAL:
* You will be going to the Taj Mahal in Agra. You will never forget it and will want to return again. Bring a good camera, and if facing the front of the Taj Mahal proper, take time to go to your left, enter a building over there where you can take photos OUT THROUGH THE WINDOWS of the adjunct for fantastic "window framed" photos of the Taj. See everything, many hours worth.
* I might mention some Hindu temples, but ... also go to a Jain Temple, a Suffi Saint tomb, Sikh Temple. I will get back.
* Have lots of rupees cash to buy wonderful crafts, art, religious items. Cash only. Don't feel sorry when you get home and didn't buy over stupid haggling of price, and keep one bag or luggage just for these.
* Bring "baby wipes" packs (in small pouch folds) to wipe your face and hands on those bus rides.
* Rickshaws are outlawed now in many cities or downtown, and while there is a lot of dark history past and present, ride one to get from here to there. It will be special, and give you a frontline view of life.
* Make friends with those on any tour. They are often just like you, and love the same things.