My answer is different than the average indian's would be. Most of my friends are through church or my husband's employer, and that's how I find out about things to do. There's also the Overseas Women's Club, but I refused to pay $40 to attend events my children aren't allowed at.
But every evening, people congregate outside our apartment building.
(It isn't usually decorated, there was something going on in the clubhouse.) Maids bring the babies down to the playground before dinner, older kids play badminton or cricket, and parents meet and talk. Every single evening. I imagine it's like this in every Indian community, no matter the class.
There are also indoor play places. It's about $2/hour for a kid to play, and they have ball pits, soft slides, ride-on toys etc. There are young women employed as attendants to watch and play with the kids, so you can leave them there. These are found in most malls and other places. We do this about once a week.
Pretty much any activity is frugal here because entry fees are so cheap, and you don't usually have to pay for children under age 5. I took both kids to a museum, including the 3D movie and spent about $2. The aquarium is $.40 entry fee. A day at the children's park with train ride and boat ride costs $4. A day at the zoo, including elephant rides and the tiger safari is $10 (for our entire family). We try to get out and do stuff like this more often than we would at home – to take advantage of the low prices, and to get out of the house.
As far as actual playdates with Indians, we've only had a few. Maggie has a new little friend we try to see once a week. They play toys and chase each other on his tricycles, just friends would at home.