Yes, Mumbai is very safe in terms of travelling. Public transport is not a problem at all. You can hail taxi / auto-rickshaws on the road anywhere. The taxi / auto-rickshaws go by the meter. All you need is the meter-to-fare tariff chart. These are available on-line at the below addresses and you can print them and keep them with you during your visit.
Taxi Services in Mumbai, Bombay Taxi Services, Mumbai Taxi Fare Chart, Taxi Helpline, Taxi Telephone Numbers, Mumbai Taxi Metre (for taxis)
Autorickshaw Services in Mumbai, Three Wheelers Service in Mumbai, Bombay Autorickshaw Services, Mumbai Autorickshaw Fare Chart, Autorickshaw Helpline, Autorickshaw Telephone Numbers, Mumbai Autorickshaw Metre (for auto-rickshaws)
There are air-conditioned cabs / call-a-cab (for which you do not need the meter-to-fair tariff chart) which are very reliable. Travelling at night too is safe. You can also hire a car from a local travel agent. If you know Hindi then it will be very easy for you to go about.
My personal opinion is Mumbai is not very touristy and there are very limited places of international tourist interest in Mumbai. But then that could be because I'm born and brought up in Mumbai and I take these places for granted. The normal tourist places are
In Mumbai
Gateway Of India
Marine Drive and Chowpatty Beach (if you are not to finicky, you must try the road-side stalls)
Malabar Hill
Mani Bhavan (where Mahatma Gandhi lived during his visits to Mumbai, now a museum)
Mahalaxmi Temple
Siddhivinayak Temple
Haji Ali Shire
RBI Museum
Jehangir Art Gallery
Prince Of Wales Museum
National Gallery Of Modern Art
Kanheri Caves and Sanjay Gandhi National Park
Near Mumbai
Ajanta & Ellora caves (these are closed for some time during the rains, so you will have to check)
Titwala
Lonavala (beautiful during the rains)
Matheran (hill station)
You can hire a car from a travel agent near the place where you stay and they can take you around. Or you can contact MTDC (Maharashtra Tourism Development Corporation) at
MAHARASHTRA TOURISM, The Official Website of Maharashtra Tourism Development Corporation, Govt. of India - Contact_us. They have “Mumbai Darshan” tours.
As for places to eat, there are many in Mumbai. If you can give and idea as to the area where you will be staying it'll be easier to suggest.
The period of your stay is during the monsoons. Mumbai rains can be very heavy. Some areas have heavy water logging. So check with your friends before making plans for the day.
Hope this helps. Enjoy your stay.