Would it be viable to visit all of these places in a single trip? How much time would it take to visit and enjoy these places? They sound like the "must see" places in India, from what I've gathered, so it would be great to visit them in a single trip, but I wonder if it would be viable in a trip of, say, 2 weeks.
It would not be possible to see all of these monuments on a single visit of two weeks duration, given that they are all scattered across the country.
Thank you all for your responses. It certainly seems that hit the nail on the head when I mentioned Goa and the fact that it's code to Mumbai makes it the best place to start. I am also quit interested in the Kashmir region seeing as I have seen some pictures and it seems like a great place to visit. Kashmir is more famous for the tensions between India and Pakistan. Has the skirmish died down altogether or is it inadvisable for someone to go there?
I would have suggested Kashmir before any other place, not because it is quintessentially Indian but simply because of the beauty of the place. The area has always been a little volatile due to the Indo-Pak tensions, but that has never stopped people from visiting. At the moment the situation is a little calmer than usual so it wouldn't be a bad time to visit. Kashmir is a quiet place, not one where you'll be shocked out of your Western comforts with too much Indian-ness at once.

The people, despite belonging to such a tension prone area, are very helpful and hospitable. If you decide to visit there, I would recommend a trek up to the Vaishno Devi Temple in Jammu, not out of any religious reasons, but the views that you get on your way down, especially if you're lucky enough to make your downward trek at dawn, is not something you will ever ever EVER forget. the mountains rise so close before your face, you would feel humbled and awed at the same time.
Other places to visit in Kashmir are Paharganj (the most beautiful village ever), Gulmarg, Sonmarg, Betaab Valley (the greenness will hurt your eyes!) and if you're up for some adventure, then head over to Leh and Ladakh
Other places in the North are the many hill stations in Himachal Pradesh and Uttarakhand...such as Manali, Kullu, Nainital, Dalhousie, Dharamshala, Shimla, etc.