It really depends what you want to see. It's probably best to come up with a list of sights a hire a taxi (much, much cheaper than Goa).

We had 3 nights in Mumbai over Christmas last year and stayed at the Residency hotel. We woke early on Christmas morning (jet lag) and walked up to CST station, then up to Crawford Market but it was too early for much to be going on. We then headed west and walked down Marine Drive. Around 11am we found somewhere for a late breakfast and then took a taxi back to our hotel for a sleep. It was nice to see the city coming to life and preparing for the day ahead. In the evening we went to Chowpatty beach and saw Indian families enjoying the food stalls and an evening beside the sea. The following day we took a taxi to Mani Bhevan, the house where Ghandi stayed when he was in Mumbai. We took a taxi to the Gateway of India and had a wander around the Colaba area and came back here later for a bit of a bar crawl and dinner.

Get a couple of guide books from your local library and see what you fancy doing. I also bought a book from Amazon called Time Out Mumbai and Goa which we used for bar and restaurant recommendations.

I loved Mumbai and would love to spend more time there.

we only had one clear day in Mumbai so used the hotel car which we knew was overpriced at £15 He took us to many of the main sights. I had been told to go to India Gate where you could pick up a car and driver with a similar itinery for a lot less.

edit to add
http://www.holidaytruths.co.uk/viewtopic.php?f=60&t=108880&start=40
an idea of what you can do in a short time

We go to Mumbai twice each winter season for probably 5 days and do much as Katie said but at a slower pace as we have more time.
You can buy the Time Out Mubai in the book shop in the Mandovi Hotel (Panjim)or the book shop round the corner (if they have it - only once been disappointed) and this is how we found Mulland (now one of our favourite places in Mumbai).
Be aware that central Mumbai (shopping bits) do not come awake until about 11 am as they don't pack up until about 9 pm in the evening and most of them sleep where they work.
Lots of things to see in Mubai - Crawford Market is OK but like markets can be a bit "samey" but the open market over the road is wonderful - lively, rushed and in your face but love it. Chor Bazar (or Thieves Market as it is know) is another wonderful place to visit - friends were very wary the first time we took them there but have visited since on their own.
Fashion Street for tops and trousers but not a lot of real fashion - Linking Road for shoes and handbags,
The Atrium if you want to buy a Bentley or anything designer and Phoenix Mills (lower Parel) like any shopping centre in the UK
Dhobi Ghats at Mahalaxmi - if you have time.
No you will have to go back and have a holiday there. WE love it and nothing will put us off visiting Mumbai

Thank you all for the reply,s and help will get the book and see what we want to do,
will get our own taxi for a few hrs both days.
thanks annx

We also love Mumbai! I would recommend a trip to Elephanta Island, a world heritage site. If you go to the Gateway of India there are boats leaving at regular intervals.
A visit to Leopolds is a must, just for people watching! Mind you the food is pretty good as well.
We also did a trip to the Dharavi slums where some of Slumdog was filmed. It is not everyone's idea of a place to visit but I have to say it was eyeopening!!! We organised this trip through Reality Tours in Mumbai and it also included a full sightseeing tour. Last year the trip cost us 800rps each and was worth every rupee!!!!
Shopping is fantastic so make sure you have lots of space in your suitcase. My best bargain on our last trip was good quality, heavy weight, cotton trousers for 50rps a pair. I didn't bother bartering!!!

Hi Squigs just checked Reality Tours looks good, think we will book it!!
fancy not bartering over 50rps thanks for all the information
annx

We did Dharavi with Reality and must say we were not impressed. We did not see much of what was promised, the guide moved us along quite quickly (think he must have had somewhere else to go) and when we asked where the pottery was we just got the head wobble and "closed" - tannery - closed school - closed. Please fill in your report with good words and into the car and he never spoke another word to us in the next hour it took to get back into central Mumbai.-
Took many emails before they sent us the promised photographs and they were disappointing - got better ones from google images.

That was a shame Waterproof. Did you complain to Reality Tours? On our trip we spent over 3 hours in Dharavi and saw everything that was on the trip details, except it was a holiday so the children weren't in school.
We received 3 sets of photographs within a month.