To explore a city, no time is enough, as i have found with my years of experience. Some hidden things will always remain. But since you do have a time-bound question, let me try and list a mix of touristy and unusual places to visit:

1. Gateway: But take the ferry to the elephanta caves. Worth the effort. Try and look it up before so you explore and know the caves better. That is the only way to see any monument, not superfluously. For a headstart, you can read this article on NearFox.
2. Then when you return, have a nice dinner at Cafe marina overlooking the sea. Rad stuff!
3. Day 2 – Visit the Nehru Centre. It is the most under rated thing in Mumbai, and a package of awesomeness!
4. Day 2 – Second half – Go to the Palladium mall – Now I am not a fan of malls, but this one is everything that Mumbai is. At night, hit the clubs there, Canvas or Tryst I forget and the experience is complete.
5. Day 3 – Take a local, ditch the taxi and go to the suburbs. There is the ever so beautiful but slightly dangerous Aksa beach and the Marve beach. Spend some time there, or go to Essel World/Water Kingdom. Make a stop at the Vipasna
6. Day 4 – back to town, visit the girgaum chowpatty. Or Juhu, if you’re on the other side. Have gola and get some photos clicked. It is a Mumbai must -do , even residents do it atleast once!
7. Visit the taj. If your budget permits have a meal there. if not, visit the asiatic library, and the kala ghoda area. absolue mumbai. buy books from fort /flora fountain. Revel. have a meal at the kala Ghoda Cafe. Life.

And above all, have a great 5 days 🙂