I am by no means a veteran, though I go to Mumbai for work a few times a year.
If you have never been to India and have the money, stay in a nice hotel. Believe me. You might even get some discounts. Look around (I'm saying this even though I'm of Indian descent and have gone to India many times before to visit family for months at a time).
Mumbai? First I would look at wikitravel for some ideas. There are some great walks, particularly in South Mumbai in and around Colaba (though it will be hot). Walking in and around Chor Bazaar can take a whole day (or years).
Take the slum tour on Sunday.
http://www.realitytoursandtravel.com/ while it was no big deal to me, my two (non-Indian) friends thought it was incredible and eye-opening.
And of course at night, eat at Bade Meya, behind the Taj Palace hotel. Everyone knows about it and it is so good.
That should be a start. Hope this helps.