Hi, 4-5 weeks in India is sufficient time to visit all the prominent places in the country. Travel would be a little hectic since its quite vast and if you are using the train, its fairly time consuming.
Based on my personal experience, I would suggest visiting the following places in India:
I assume you would land in Delhi, so do visit all the prominent places here. From Delhi you should take the train to Haridwar (avoid the bus or cab) the train is a lot more comfortable. You'd get to Haridwar in about 4-5 hours by train from Delhi.
From Haridwar take a cab/taxi or bus to Rishikesh. Cab would be around 700-800 INR whereas a bus fare would be about 50 INR. I would suggest taking accommodation only at Rishikesh and spending more quality time there. Haridwar is far more touristy and less clean. Plus there is not much to see here, so half a day of site seeing would cover everything possible.
Rishikesh has a lot of yoga ashrams, river rafting, trekking and a lot of other adventure sports you could indulge in. Else there are meditation centers and temples that you can visit.
If you are looking at including wildlife in your trip - then travel from Rishikesh to Jim Corbett National Park. Its a tiger santuaary plus you'd get to see a lot of other wild life too there. You can goolge it up or check blogs written on this site.
You'd have to come back to Delhi (from Rishikesh or Corbett) and travel towards Agra. Agra is about 4 hours by train from Delhi. You can see the Taj Mahal - ofcourse - but also do visit Agra Fort, Sikandara and Fatehpur Sikri. I personally think that Fatehpur Sikri is the most beautiful architecture.
From Agra travel to Jaipur (another 4-5 hour journey) and from Jaipur to Udaipur. I would ask you to skip Jodhpur and Jaisalmar on your first trip bcoz though they are really beautiful and worth a visit, if you spend a lot of time in Rajasthan, you will miss out on other nice places.
You could instead travel from Udaipur to Khajuraho and Orchha in Madhya Pradesh and from there travel to Mumbai. After spending time in Mumbai you can go to Goa.
I do feel you'd be missing out on a nice place if you skip Goa - Goa is renowned for its beaches. Its full of beach huts and shacks where you get loads of beer and seafood and can party through the night. If any of this excites you, do make the trip else free to drop it from your plan.
If you do visit Goa, travel from Goa to Kerla, to the backwaters in Alleppey. Some other destinations worth visiting are Munnar, Thekkady and Kumarakom. You must also visit Kanyakumari - southernmost tip of the Indian Peninsula. You can actually see tThe confluence of three ocean bodies - the Bay of Bengal, the Indian Ocean, and the Arabian Sea here.
From Kanyakumari you can travel to Chennai, visit Mahabalipuram and Pondicherry and since Chennai has an international airport, you can fly back from Chennai instead of traveling back to Delhi to board your flight.
If you do need any specific information, do write back.
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