I finally got the chance to explore Bombay on my second day in India. I started by asking the hotel to get me a taxi to go to a market downtown. The hotel worker asked me, "370 rupees?" and I figured the hotel wasn't trying to rip me off, so I said okay. But, in retrospect, I think the taxi drivers have some sort of side deal with the doormen. The driver managed to convince me that where I wanted to go wasn't worthwhile (it really wasn't) and that I should get a whole day tour from him for 1200 rupees. I asked him if that included time if I wanted to stay out for dinner. He said it was no problem, and that there was "No charge for waiting."
So, with that, he took me to the handful of tourist sites in Bombay. Gateway of India - check. Prince of Wales Museum - check. Marine Drive to see the "nice" office buildings - check. Chowpatty Beach, a stretch of sand located downtown - check. Hanging Gardens, a small park located on a hill overlooking Bombay - check. At around 4 PM, despite my earlier desire to stay out and eat dinner, I couldn't take the dirtiness and commotion any more. I told my driver to take me back to my hotel, where I took a shower and ate dinner.
Here's my quick summary of Bombay. It may be interesting to see as a commercial hub and in comparison to other cities worldwide, but I found it even more lacking in tourist locations than Shanghai. It lacks the tall skyscrapers of Shanghai, the cultural attractions of New York, and the historical context of London. It's worth a quick stopover, but one day is probably all you need.