Mumbai is the new name, which is related to the older pre-colonial name. A while back a fairly nationalist government changed a lot of names to try to slough off some colonial heritage.
I think your first step would be to grab a copy of the Lonely Planet and get a better idea of what you are looking at. While the Lonely Planet has it’s ups and downs, the India edition is pretty helpful, covers a good range, and has really helpful history/culture/reading recommendation sections. It goes through the ins and outs of getting from place to place, etc. A whole lot of research has gone into the book, and it’s a good place to start thinking about your trip. They even have recommended itineraries.
I spent three months budget (less then $10 a day- for everything) backpacking all over India and it’s one of the best things I’ve ever done. India is an incredible, unbelievable, just breathtaking place. That said, it’s not for everyone. It’s the most pain-in-the-butt place to travel in the world, with the most persistant touts, insane scams, and dangerous chances to become a crime or accident victim. In both the good and the bad, it is a really, really intense place (unless you spend a ton of money on luxury touring that isolates you from the actual country.)
If you are going to have a good time, you are going to need to be patient, have a good sense of humor, and do your best not to get worked up about hassles- because there will be plenty of hassles. The only way to enjoy yourself is to learn to let stuff go and realize you are going to get ripped off now and then and there is nothing you can do about it, and you gotta just shrug it off, move on, have fun, and stop stewing over the fifteen cents the cab driver overcharged you.
I would stay as long as you can. Getting there is the big price. Once you are there, depending on what you are willing to put up with, you can live pretty inexpensively. It was quite a few years ago, but I hung out in Mysore for a while sharing a 1.75 hotel room and eating .50 all-you-can-eat meals and $.10 street snacks. Even staying at slightly nicer guesthouses- which often have a lot more charm than the countless generic hotels- can be pretty cheap. The Lonely Planet has a good range of recommendations, and I had lots of fun at the mid-low range places described as “slightly funky but charming” or “simple but clean.” In my experience the faceless mid-range hotels are not much better than the guest house, and the guesthouses are a lot more fun and hanging out with the owners is an easy way to get some local perspective. That said, there are honest-to-god luxury hotels and they are a great value.
India is HUGE and there is only so much you are going to be able to experience in a few weeks. I’d warn you against trying to do too much. Think of it like “America,” you can’t “see” America in a few weeks, but you can see a part of it. In a month you could probably explore the highlights of two regions- maybe the ancient holy city of Varanasi and Agra to the North, and maybe make it to bizarre landscape and ruins of Hampi or some of the beaches (I liked Kerala better than Goa, but that’s just me) to the South. Don’t linger more than a couple days in Mumbai- it’s expensive and a pain in the butt. There are much better places you can go. I also recommend Rajasthan to anyone and everyone. You could easily spend the majority of your time there exploring it’s storied city and desert.
Wherever you end up, the “attractions” are petty repetitive- almost every city will have a temple, a holy mountain, a market, a palace. Besides the Taj Mahal, it’s not so much about “must see” sightseeing. It’s more about people watching at temples, exploring old cities, chatting with people in parks and generally soaking up the atmosphere. The attractions are really just an excuse to get out of your hotel and explore. So it’s better to linger in a place a few days so you can really get to know it.
India has a great train system and a terrifying but convenient bus system. You should ride at least one overnight non-AC sleeper train for the experience of it. That said, a lot of discount airlines have popped up and I believe internal travel is pretty reasonable these days. India has a lot of people, so there is pretty much always some way to get from one place to another.
Here is a journal of my adventures- you might want to start from the end and read it chronologically.
http://jeninindia.livejournal.com/
I’ll check back here and also feel free to PM me if you have any questions. I did this in 2004 so no doubt a lot has changed since then, but I covered a pretty wide range and travelled pretty close to the ground, so I’ve got plenty to share!