Hi,
India is indeed a huge country and every state has its own unique charm. If you're going to Delhi you cannot miss a trip to the Taj Mahal!
Goa is a great place to unwind near the beaches especially during December as its the peak season and there is a festive spirit in the air which is great! I guess one of the best places that I would recommend in Goa is Baga beach but there are several other beaches that you could check out as well. There is really good sea food and lots of water sports that one can indulge in while in Goa. As for the transport in Goa you could rent a two wheeler and drive around which works out quite cheap!
Apart from these cities you could come to Bangalore which is a nice city located in Southern part of India with awesome weather and there is a lot to do here. There are several cultural programs that take place often (in Chowdiah Hall) so you can get a taste of the local culture which include traditional Indian dances or music. Sometimes there are even Jazz concerts that take place here. There are also many restaurants with different cuisines right from typical south Indian to north Indian to Chinese to Italian food! Bangalore is also known for its unique pub culture which one should definitely check out.
Kerala is another place that you could check out, its also located in the Southern part of India and is known for its lovely and serene backwaters apart from the other attractions. A place that I would recommend in Kerala is Kovalam which is a picture perfect tropical beach. Some unique features of Kovalam (or for that matter Kerala in general) are the Ayurvedic massage parlors and yoga centers apart from the delicious seafood and excellent accommodation!
As for the mode of traveling when you are in the North you could take the local trains to travel to other cities as it would be a lot cheaper but when you decide to come to Goa or the South you could take a flight (as it would save time) and then you could travel either by train or bus to get to places which are located in and around the South.
Hope this information was helpful to you!
Maybe when you're sure of which places you are going to visit you could get more help