Hiya, i spent about 2 1/2 months there a couple of years ago travelling around solo, and still barely scratched the surface!
But still have a few tips that may help:
Fly into Mumbai if you can as its a better start point than Delhi to ease yourself into the country.
Stay somewhere fairly central such as Colaba for a couple of days and absorb the sights - and smells...
Do the usual tourist stuff whilst there such as the Gate of India, Cafe Mondagar (sp?), Cafe Leopold, Victoria terminus, Elephanta island etc etc. If you're feeling up to it then Colaba docks are fascinating early in the morning when they unload the days catch - it bloody stinks mind.
Met a few people whilst there who had been asked to star in a Bollywood film and they said it was good fun. Got asked myself, but i was on my way to catch a train at the time and sadly didn't get the chance to make my debut on the silver screen.
From Mumbai have a look at the maps and have a think about what you are interested in, its a huge country and the trains are slow, so you will have prioritise the places you really want to get to.
If i was doing it in your timeframe i'd get the train out of Mumbai and head north toward the Punjab and Amritsar, the Golden Temple is sublime at both sunset and sunrise. You can get a taxi from there to Attari for the Pakistan border closing ceremony in the evening which is a must see IMO.
From there i would continue to head north and go to either Shimla or Kashmir for a few days and see some of the Himalaya. Dal Lake ( Srinagar ) is worth looking at if you do get to Kashmir. Although the roads getting there will send you grey... ( the train doesn't get you all the way )
If you can make the time the Dalai Llama's monastary in Mcleod Ganj ( Himachel Pradesh ) its well worth visiting and the general vibe in Dharamsla is fairly chilled for India. However its a bit time consuming to get there.
I wouldn't spend too much time on the Golden Triangle, the sights are must see, but a day at each should be enough any more and the constant hassle from touts etc may well drive you crazy. If you schedule it right then leave these till last and see them on your way to Delhi to fly home.
However if you do choose to start off with this, then maybe continue east and go to Varanasi, and then Khajuraho - seeing what is essentially porn carved in stone in such an otherwise conservative country, even when it comes to just holding hands is surreal!
Trains are definately the way to do India, they are slow paced, stopping frequently for livestock etc but are a wonderful chance to meet Indians, who tend to be very friendly and curious when you travel with them. Dont be afraid to eat the on train meals either, they tend to be rather good!
I found train class AC2 ( second class AC ) is the best compromise between comfort and price. As a bonus you'll find travelling in this class is more likely to sit you with Indians who speak good English.
The protocol of ticket booking has changed a little recently i believe, but usually there will be a tourist quota for each train. But make sure to book any onward journeys early as these quota's will sell out. Ignore any tout who grabs you outside the station, suggesting that you cant buy a ticket - they will probably be scamming you, found this out the hard way at Delhi!
Let yourself be absorbed by India and go with the flow. Be prepared to change your itinerary based on the whim of train availability and other interests that present themselves. And if you do get scammed along the way remember that the value of the scam is normally pretty small, so not worth stressing over!
This rather long winded reply should give you an indication that there is a huge amount to see there, i've not mentioned Goa, Kerala, any of the eastern states etc etc etc!
Oh and as a final point, if you've not read it already then "Shantaram" by Gregory David Roberts is well worth picking up before you go for reading on the plain / train.
You've started thinking about going back myself now dammit, so off now to check out Air India flight prices..... Any airline that has Old Spice in the bathrooms whilst serving G+T's and Bombay mix on route gets my business!