India is vast and it is not possible to visit every part. And every corner in India has its own culture.Its a very diverse country and so its called "Unity in Diversity!" You might need years to just experience it. But if you have limited time this should help]
1) Southern India - Land in Mumbai enjoy couple of days here (Ajanta and Ellora Caves don't forget) and then go to Goa, then go to Karnataka (Mysore and touch base Bangalore), fly to Kerala (gods own country),touchbase Munnar and stay on house boat for a day and experience it, from there go to Kanyakumari, Pondicherry and Chennai and then to Hyderabad[keep atleast couple of days here][try Tirupati if you can. This will be another experience]. This should take you atleast 3-4 weeks with heavy travelling.
2) Northern- Western India - All historic places in Delhi,[Jim Corbett National Park. I won't miss this but if you don't wish leave it] Jammu & Kasmir, Chandigarh, Rajasthan(Jaipur, Udaipur, Jaisalmer), Gujarat(don't miss the rann of Kutch) Atleast 3weeks
3) Eastern India - Land in Kolkata and experience kolkata. Historic place, Amazing food, Sweets etc. Then travel to Orissa. Puri, lots of temples from olden times, lakes etc. [Eastern part of india is very very beautiful. There are some amazing spots in Assam and Nagaland but check the current safety situation before planning.] [Atleast 2 weeks]
4) Andaman is beautiful and make a short 3 to 4days trip to enjoy it.
Whatever I write its very less. I cannot cover everything. It's humanly not possible. Please plan early and do research about each section before you go. In India Tripadvisor will have lot of updates and reviews. For food try zomato or other online sites to do research. For booking tickets use online websites to do research before going to agents. Some good website like cleartrip, irctc etc. You get trains for every place. Its cheap but takes time.
To experience India you need to eat. Eat everywhere you go. Take medicine on time and eat and eat. Don't miss anything. There is awesomeness in every corner and you will find new things every where.
If you have any Indian friend, brainwash :P him to come with you. Go to his home. Have some home made food. It will add more joy and you will get more details about the culture.
[And if you are one of those who likes things to be taken care of then there are reliable websites like Yatra.com, makemytrip, thomascook etc who can create a package. Tell them these options and they should take care. I have very good experience with Yatra.]
All the best and enjoy the nirvana.