Colaba causeway.. beside gateway of India, take a ferry from there to elephanta caves, then marine drive, girgaon chowpatty. take a ride on bandra worli sea link, sunset at Bandra bandstand beside SRK's house. u can also go to borivli beside essel world for the vipassana pagoda, amazing place .
Bombay port trust garden
gateway of India
nariman point hanging gardens
walkeshwar viewing point
borivali national park
In particular Mumbai for view of sea,
if u are a shopoholic person
visit fashion street
if looking a bit beyond mumbai,
raigad, alibag, Ratnagiri etc
To see victorian time Mumbai and posh Mumbai, go to churchgate station. Go to CST station, colaba causeway, marine drive . And for Bollywood darshan, go to goregaon east, go to film city. For beaches, go to juhu beach, although it's a clean beach but food is good there. You may see some celebrities there too. But for much beautiful beach experience, go to Gorai beach. it's a typical Marathi locality beach. you will get a Goan type feel also. it's a clean and secluded beach. For shopping , do visit Linkroad , Alko road, crowford market for shopping.