Traffic in Bombay is chaotic, but relatively disciplined compared to other cities in India. There are NO donkeys, elephants and cows on the streets of Bombay. They were banned a long time ago. Plenty of stray dogs though. I wouldn’t reccomend driving yourself, unless you’re used to our kind of traffic (New York and Boston is childs play for the seasoned Bombay driver).
Taxis and autorickshaws fares are calculated by meter, and a rate card is available with the driver. Always make it clear that you will only pay the metered fare only, and ask for the rate card before paying. Late at night, expect to pay upto 50% more than the metered fare though. You will almost always be taken for a ride by the cab/rickshaw driver (taking roundabout routes), so it would be best to get to know your regular route as soon as possible. I hope, for your sake, that your company will provide you with chauffered transportation!
Bombay’s international airport is probably the best in the country, and Zcrysis has described it fairly accurately. It has a long way to go by international standards, but it’s reasonably ok. The domestic terminal is a far sight better, so flying out to Madras will be much more comfortable. Domestic airlines provide very good service, are pretty much always on time, and yes, are a bit more expensive than the state-run Indian Airlines. Take my advice and never fly Indian Airlines - they have absolutely no concept of punctuality. Jet Airways and Sahara Airlines are the best. There is a no frills airline that has recently started, but they don’t have Bombay-Madras flights as yet. You’d have to fly to Bangalore first, and then to Madras, which would work out to the same cost as a direct flight on the regular airlines.
So much for transport in Bombay. I wish I could tell you something about Madras, but it would all be guesstimation, at best, seeing as I’ve never been there myself.