You can start these activities under one umbrella, name it like HR Connect or Fun Zone.
Take manager's approval when they can spare team members for these activities, maybe initially every 3rd Friday of the month.
To create a sense of being part of the team, form a group which creates the buzz. You can start with simple activities like Antakshari, company quiz, song and dance, laughter challenge, acting/mimicry, etc. Then, if budgets permit, go for showing motivating movies, competitions like rangoli, bouquet making, carom, chess, etc.
You can also include celebrating festivals and company days.
Once you start these activities, suggestions will come from within the team. Make sure the excitement is shared by all; HR alone should not do everything.