DH and I have just returned from an 18 month assignment in Bangalore. Overall we enjoyed our time there but now we're back in London we don't miss the city one bit iyswim. Although we had a great lifestyle there in many ways, it was a difficult city to live in and you need to be prepared for that if you do go.
First thing I'd say is make sure the finances stack up. A lot of people assume that India is a cheap place to live but this isn't necessarily so. For example, our rent in bangalore was similar to the amount we paid in London. We got a bigger place for it but when we were looking at properties there wasn't anything we liked that was cheaper. When I say anything we liked, I am talking about fairly basic things like western style bathrooms and kitchen with mod cons and adequate security. Also international schools are costly so make sure they will be paid for as part of your package. Your expat package should ideally also include a driver and car as you won't manage without one day to day.
In terms of lifestyle, most people with families tend to favour the gated communities in Whitefields. They are generally modern developments with a number of villas on a secure site, usually with shared facilities like outdoor pool, gym, club house and sometimes tennis courts etc. If you live in one of these, you will probably know most of your neighbours (who will be other expats or wealthy Indians) and your children will be able to play outside with minimal supervision. Most people living in these sorts of developments wouldn't worry too much about locking their doors if they are in or around the house. You will also probably have a maid and driver around who will keep an eye on the property and provide a bit of extra security. The main security risk is petty theft so for example if you have workmen you don't know in, you or your maid should keep an eye on them as this is usually when thefts occur.
With regards to schools, there are two or three international schools which most of the expats favour. Our DS isn't of school age so I have no direct experience but everyone we knew was very happy with the schooling.
I notice that you said that your children currently have a outdoorsy lifestyle and wanted to warn you that this was one of the things I really missed in India. There is very little green open space and the streets outside the gated communities are dirty and unsafe (lack of pavements, bad traffic, no safe places to cross etc) so it wasn't really possible to go for a walk. I think without exception if you asked most of my friends there what they missed most about home, after family and friends, they would all say going for a walk or going to the park.
On the plus side, Bangalore has the most amazing restaurants, with a huge variety of different cuisines available and very reasonable prices. We ate very well while we were there. The climate is fantastic, usually between 25 and 30C, with very little rainfall (there tend to be some short showers during rainy season. And there are pretty good links to the rest of Asia for holidays. Most people who want a break from India will either book a weekend in Singapore or a few days in Thailand. We also had short breaks in Hong Kong and the Maldives while we were there. Some expat packages will include an allowance for flights home, which you can either use to go home or instead use to travel in Asia like we did.
I hope that is helpful, let me know if you have any other questions.