I travel to Bangalore on business about once per year. My local sales guys always try to take me to a new temple or palace when I go including traveling to Mysore.
In Bangalore there is of course numerous Hindu temples and even a Hare Krishna temple. ISKCON Sri Radha Krishna-Chandra Temple is a beautiful temple to visit but they will not allow you to take your camera gear in. I never have enough time to visit some of the natural parks but you should be able to get a driver and car for fairly inexpensive hourly or daily fares.
In Mysore you should definitely visit Mysore Maharajah's Palace. It is beautiful, huge and an amazing piece of architecture and history. You should also visit Sri Chamundeshwari Temple which is one of the highest temples in southern India. Enjoy all of the good food in the area!!
Kabini Wildlife Sanctuary is worth a visit for birds and mammals but it might be a bit far for you - 95kms from Mysore - and really requires an overnight stay to make the most of it.
I love Bangalore for street photography - perfect for your m5 if you are taking it. Commercial Street is paradise for this, with so much happening and very safe.