Vivek Express: ऊंचे नीचे रास्ते और मंज़िल तेरी दूर.

If ya wanna chill and explore Mumbai at your own pace instead of rushing with those Mumbai tours, peep this. (Started from CST).
Day 1. South Mumbai (Gateway of India, Marine drive, Museum, Elephanta caves)
Right outside CST, there's a BEST (Mumbai transport service) bus stand. From there:
Bus no. 111 - Headin' to Gateway of India (final stop), comes every 5-10 mins. Fare - Rs 5.
From Gateway, catch a ferry to Elephanta caves for about 150-200 bucks (round trip) dependin' on the boat you choose. There's also ferries goin' to Alibaug.
Once you've checked these places out, head back out the Gateway from where you came. Go straight and you'll hit SP Mukherjee Chowk. Cross over to the other side and you'll be in front of Chhatrapati Shahuji Maharaj Museum (pretty cool spot).
Then grab any bus (I suggest 108, 111) back to CST.
In the evening, hop on Bus no. 108 to Kamla Nehru Park (fare - Rs 15). Check out the hanging gardens, hop back on the same bus to Chowpatty & Marine Drive. Have a blast in the evening and head back to CST on bus no. 108 or head to your hotel, your call.
DAY 2 (Siddhivinayak, Bandra)
Better go early mornin' to Siddhivinayak to dodge the long line. I usually go around 6 AM.
Take a local train from CST and get off at Dadar. Use the FOB and head to Dadar West. From the Flower Market (right as you step down from FOB) you can find shared cabs to Siddhivinayak (fare Rs. 10).
If you go early morning, head back to your hotel after darshan & wait till around 10:30-11 AM (avoid the rush hour). At 11 AM, catch a local (specifically Bandra Terminating local) from CST. Once you get to Bandra, move towards the west side. You can find shared autos and Bus no. 211 (final stop is Bandra Fort/Father Agnel Ashram, get a ticket till there, fare - Rs 5). Go see Band Stand, Bandra Fort, peep the Sea Link, film stars' pad. Head back to CST and relax. In the evening, you can catch a sunset at Marine Drive or do some shopping in South Bombay (up to you).
Day 3 (Juhu Beach, Film City, Haji Ali)
Get on bus no. 28 from CST (outside the bus stand), will drop you at Juhu Beach. Enjoy the mornin' (you might spot some celebs there). Take a local train to Goregaon from Vile Parle station, then take an auto to Film City. There's a 2 hr guided tour (rs 600) to see the famous spots. Come back to Goregaon and catch a CST local.
In the evening, hop on bus no. 28 and head to Haji Ali Dargah and if you have time, you can also stop at Cumbala Hill Post Office bus stop (for the National Museum of Indian Cinema) on the way back.
These are the main spots I've hit. Plenty more places you can check out with Mumbai's public transport system.

Agar tu Mumbai ko apne pace se dekhna chahta hai aur jaldi mein nahi, tab yeh rasta hai. (Maine CST ko base banaya hai).
Day 1. South Mumbai (Gateway of India, Marine Drive, Musuem, Elephanta Caves)
CST ke bahar, BEST (Mumbai transport service) bus stand hai. Wahan se:
Bus no. 111 - Gateway of India ke liye (last stop), har 5-10 mins mein available hai. Fare - Rs 5.
Gateway se elephanta caves ke liye ferry mil jayegi jiska cost boat/deck ke type ke hisab se 150-200 (round trip) hai. Alibaug ke liye bhi ferry service hai.
Inhe explore karne ke baad, gateway se wapas exit karna jahan se enter kiya tha. Seedha chal. SP Mukherjee Chowk tak pahunch jayoge. Dusri taraf cross karke, Chhatrapati Shahu Ji Maharaj Museum ke samne aaoge. (Dekhne layak).
Phir koi bhi bus le lo (preferably 108, 111) CST ke liye.
Shaam mein, Kamla Nehru Park ke liye Bus no. 108 le lo (fare - Rs 15). Hanging Gardens dekho aur fir wahi bus le lo chowpatty aur Marine Drive ke liye. Shaam ka aanand leke wapas CST ya hotel ja sakte ho jaise marzi ho.
DAY 2 (Siddhivinayak, Bandra)
Lambi line se bachne ke liye subah jaldi darshan prefer karo. Main usually 6 baje darshan kar leta hoon.
CST se local pakdo aur Dadar pe utro. FOB leke Dadar west ki taraf niklo. Flower market se (jab FOB se utarte ho) siddhivinayak ke liye shared cabs mil jayegi (fare Rs. 10).
Agar subah jaldi gaye ho, toh darshan ke baad hotel wapas jaake 10:30-11 baje tak wait karo (peak hours se bachne ke liye). 11 baje CST se local pakdo (preferably Bandra Terminating local) aur Bandra pahunchne ke baad phir se west ki taraf exit karo. Shared autos milenge aur Bus no. 211 bhi milegi (last stop bandra fort/father agnel ashram tak ticket lo, fare - Rs 5). Band Stand, Bandra Fort, Sea Link, Film stars ke banglow/flats dekho. Fir CST wapas aa jao aur araam karo. Shaam ko sunset dekhne ke liye Marine Drive ja sakte ho ya south Bombay mein shopping bhi kar sakte ho (apni pasand).
Day 3 (Juhu Beach, Film City, Haji Ali)
CST se Bus no. 28 leke jao (bus stand ke bahar), Juhu Beach pe pahunch jayega. Subah ka maza lo (kuch film stars bhi dikhe sakte hain). Vile Parle station se Goregaon ke liye local leke jao aur fir filmcity ke liye auto pakdo. Wahan 2 ghante ka guided tour hai (Rs 600) jo tumhe famous spots dikhayega. Goregaon wapas aake CST local lelo.
Shaam ko, bus no. 28 pakdo aur Haji Ali Dargah jaao aur agar time hai toh pehle Cumbala Hill Post Office Bus Stop (National Museum of Indian Cinema ke liye) jaake dargah dekh lo.
Ye bade spots hain jo maine cover kiye hain. Aur bhi jagah hai jo Mumbai ke public transport system se accessible hain.

Thank you for providing such a comprehensive and detailed itinerary for exploring Mumbai at one's own pace. Your suggestions for places to visit and the transportation options to get there are very helpful. It's great that you have included information about various modes of transportation like buses and trains, as well as shared cabs and autos. This will allow visitors to choose the most convenient and cost-effective options based on their preferences. I'm sure this will be a valuable resource for anyone planning to explore Mumbai. Well done!