Hi Desperado and welcome. First of all I agree with everything KK says - pick North or South, but don't try to do both in one trip. We also try to do a lot (too much) on our trips ranging from 21 - 26 days. We're about to take our fourth and recognise that even at our frantic pace, we will need at least three more to just scrape the surface of all we want to see in India. When we're done, we will revisit the places we loved most, take longer and rediscover India at a deeper level. Our difficulty is that there are too many choices and our interests are wide-ranging - we go for culture and history in a big way like yourself, but find that it's often the slightly off-beat things that we enjoy the most. So you have time to do some research and pick a corner of India where there are things you MUST see NOW and plan a trip around a few of those, not too far apart, with interesting journeys between. Be prepared to book things ahead if you want to cover a lot of ground. Or, as we do, pay an agent to do the groundwork for you, so you can make the most of your precious time.
I can give you a few ideas for places you should see near/around some of the cities you name:
Mumbai
Elephanta Island obviously. Get the train out to Aurangabad to see the Ajanta caves (renaissance quality paintings from the 2nd to 7th centuries miraculously preserved in a glorious setting) and Ellora (rock-cut caves and temples, hewn from a cliff - the phenomenal Kailasha temple - the worlds largest monolithic building). We wish we'd know before we went that this city is worth at lease three days, to see the Daulatabad Fort, the Bib-ka-Muqhara ("mini Taj"), and the local caves. But we didn't, so we missed them.
Also, go further north to Mandhu (near Indore). Unspoilt village among the ruins of a Mughal city, worth so much more than the day and a half we gave it. It's not all that far from Udaipur either.
New Delhi
What a city - more history and culture than you can shake a stick at, Jyotirmoy is your best guide here. Humayun's tomb and the Qtab complex are essential, Jama Mashid and the Red Fort are too iconic not to be missed either. But if you're pushed for time, you could miss going inside the later if you see the Agra Fort instead.
Agra
As above, the Fort is glorious and in better condition than the one in Delhi. Also see the Itmad-ud-Dullah tomb - exquisite. If you're in that area, you have to go to Fatephur Sikri - Akbar's masterpiece/folly. It's only another hour or so down the road to Bharatpur/Keoladeo National Park. And that's not far from Deeg which we haven't seen, but certainly will next time we're in that region. Alternatively, head the other way to Orcha, Jhansi - they look great, but again, that's another trip for us.
Goa
Unless you're a beach bum, a couple of days is enough to see the main historical sites (the cathedrals and churches and the pretty old town) But about a day by car/bus would get you to Hampi in Karnataka, needs three days really - we did it in two.....

There are some great threads on this place somewhere, it's vast and astonishing and I've never met anyone who wasn't blown away by it.
Head further north to Badami, Aihole and Pattardakal - some very early temples.
Mysore
Lovely place. You're not far from Somnathpur temple - a little jewel, and the other Hoysala temples - Belur and Halebid are about a day away. Ooty and Coonor are just a few hours up the hill if you fancy a cooler climate and some vestiges of the Raj.
Most of the rest that you mention are on our itinerary for this year! Check out my thread on that - you might get some ideas - as I did! I'm still learning from these guys too......
I won't mention Mahaballipuram, Tanjore, Kanchipuram etc, because that would just confuse you even further....

nor
Gujarat or lovely Eastern Madhya Pradesh, as they will also distract you....

good luck trying to decide.
Edited by torryquine, 21 January 2009 - 02:30 AM.