Mumbai is the financial capital of india...
As such Mumbai is an expensive place to live in but it has something for everyone...
If you want to stay for really really cheap, i would suggest, go to Mumbai Central and right outside the train station you will find a number of people handing out flyers about hotel rooms for 300-400 rupees a day! I'm not guaranteeing quality though...
Food is not generally an issue in bombay coz there's cuisine of every kind! You can get good cheap food from a lot of nearby places...you must try the local stuff - panipuri, pao bhaji, vada pao and watermelon juice!
Some of my favourite food joints (that are not too expensive either):1. Cafe Churchll, Colaba (Continental food. Meal for 2 between Rs.300-400)
2. 5 spice, Fort and Bandra (Chinese food. Meal for 2 for Rs. 250-350)
3. Red Box, Bandra (Several cuisines. Meal for 2 for Rs. 300-500)
4. Shiv Sagar, Juhu (North Indian, South Indian. Meal for 2 for Rs. 150-Rs.250)
5. Rajdhani, Inorbit Mall, Malad (North Indian. Meal for 2 for Rs. 250-300)
6. Pop Tate's, Malad/Seven Bungalows(andheri), Andheri east (Several cuisines. Meal for 2 for Rs. 200-400)
Some places you must go to in Bombay:1. Prithvi Theatre, Juhu
www.prithvitheatre.org/2. Borivali National Park, Borivali (Go to Kaner caves as well as the jain temple)
3. Manori (An island near Malad. You need to get off at Malad and take an auto for gorai beach from where you need to take a steamer to manori. There you will find hotels and shacks you can spend the day at or stay overnight as well)
Traveling around bombay:1. Bombay is well connected to the rest of india by train, buses and air. For train bookings, go to
www.cleartrip.com2. For buses, you will get plenty of buses on the spot if you go to dadar station and get off on the east/central side. Buses go to pune every few minutes and cost about Rs. 180.
3. Number of buses go to goa and other destinations from there as well.
4. In some seasons, you can take the catamaran from Gateway of india to alibaug.
There are a number of beaches around bombay like kashid, kihim, murud janjira and its not too expensive to stay in those. You can pretty much go backpacking through them on a low budget!