Hey Jason, The nearby places to visit nearby Mumbai in June would be :
1) Alibaug 2)Kalamb Beach (Nallasopara) You can even go on a trek. Kalsubai would be the best option. You can visit Karnala fort.
In june there are rainy season is there so you can visit mumbai an feel the beauty of mumbai
u can go to marin lines, worli cfs to enjoy rain
In mumbai there are good waterfalls so u can go there an enjoy