Saputara is great choice, Gujarat's only hill station not far from Mumbai. You can enjoy Rain, greenery, Waterfalls, lake, jungle, tribal art, view points in Saputara
You can definitely visit Panchmarhi,Madhya Pradesh. It's a beautiful place to visit in monsoon. I myself visited that place in monsoon and it will be near to your lication aswell as in budget. you can hire a traveller open jeep, they will show you the various places, including waterfall, caves, temples and lush green forest. Being a cantonment area, it is much greener and weather is just awesome.