Mumbai is a nice city to explore on your own and there's no need for a guide as such. If you want to meet backpackers then the Colaba area is full of them and there are many places like the Leopold's Cafe and Mondegar Cafe where you will find a lot of the travellers enjoying some fantastic breakfast early in the morning or some beers at lunch or dinner. You might just be lucky to be asked if you wish to act as an extra in a Bollywood movie too!!!
If you are on your own, just visit the Gateway of India and you will also get a nice view from the top if you go to any of the higher floors of the Taj Mahal Hotel. If you want to visit the Elephanta island for the famous caves then take a boat from the Gateway and it will take you about an hour to get there. The tickets can be bought in front of the Gateway from one of the booths selling them. There is a free guide inside the caves complex from morning to afternoon (just inquire with the security guard for his/her whereabouts). After you get back from the islands you can carry on walking towards the Regal Theatre and see the Museum as well as the other places for art like the National Gallery of Modern Art and the Jehangir Art Gallery for some interesting art exhibitions. Then carry on towards the other side of the University of Mumbai buildings and see the Rajabai Clock Tower, Library and Convocation Hall. Also see the High Court of Mumbai buildings. Then if you look across there will be a game of cricket going on in one of the famous playing grounds (maidans) called the Oval Maidan. Across the street is the Eros theatre and opposite that is the Churchgate railway station and in front of that is the Western Railway Headquarters as well as the Tourism office. You can then walk down to the Marine Drive promenade and take a long walk towards Chowpatty beach for a snack of Bhelpuri, Pav Bhaji and also some Indian ice cream called Kulfi and another sweet dish called Rabdi. There are some other nice places to see in between like the Flora Fountain and Hutatma Chowk (Martyr's Square) and the St. Thomas' Cathedral, which is very beautiful with a lot of nice sculptures inside it. Nearby is the Bombay Stock Exchange building and also the Jewish David Elaiyahoo Synagogue and the David Sassoon Library as well as the Elphinstone College buildings opposite it. The Town Hall building with the Horniman Circle and gardens is also worth seeing behind the cathedral. Visit the only UNESCO World Heritage Site in Mumbai city - the CST station building besides the other site in Mumbai which is at the Elephanta Caves on the island by the same name. There is the General Post Office building which is also worth seeing and a small part of the original Fort Wall next to the St. George's Hospital. You can visit Mani Bhavan (the house in which Mahatma Gandhi lived in Mumbai) which is now a museum devoted to his work and life as the father of the nation. Nearby is the F. D. Alpaiwalla Museum which has a lot of Iranian artifacts housed in the Khareghat Colony Hall and if you climb the steps leading up to the Parsi Towers of Silence and Hanging Gardens from the outside, then you will be able to get a good view of the Queen's Necklace from above. On the way down, you can visit a very elaborately carved Jain temple and then come down towards the Governor of Maharashtra's bungalow before reaching Chowpatty beach. You may carry on towards the Mahalakshmi temple and the Haji Ali shrine in the middle of the sea followed by a trip to see the Dhobi Ghat which is from the bridge over Mahalakshmi station.
If you are interested in shopping, then depending on what you want to shop for there are many specific markets like Zaveri Bazaar for gold and silver jewellery, Mangaldas Market for clothes, Chor Bazaar for antiques (and not so old stuff too!!!), Bandra Linking Road for lots of designer shops. Lots of other shopping arcades and malls have recently opened up in many areas so you can just walk into one for a quick look and buy whatever catches your fancy. There is a small shop called Avante just opposite the Regal Cinema at Colaba where you will get the best bargains in handicrafts, souvenirs, etc at very reasonable and cheap prices. Also nearby on the Phirozshah Mehta Road, there is a shop called Bombay Store which has a lot of handicrafts and good Indian teas as well for sale in attractive packaging which is another excellent gift idea.
