Need help regarding accomodation.

I (31 M) will be joining IIT as a post-doc scholar on 1st April. While they are not providing accomodation within campus, I will be getting around 21k as an HRA (I am ready to add a few thousands, if it leads to better options). I will be coming from Delhi and have zero knowledge about the rent rates of the city, specially in nearby areas of IIT. So, I have a few queries regarding it,

1. What kind of options should I expect in areas nearby to IIT, within range?

2. What are the general do's/dont's while looking for places to rent in Mumbai?

3. What are the do's/don'ts in general for someone new coming to the city?

It would be a great help if you could just ease my anxiety of joining the place.

Would love to hear more suggestions/tips that you might have.

Thank you.