Hi Vaidehi,
Welcome to the US and to North Carolina!!
Even though I stay in Atlanta (a few hours away from Charlotte), but I've been there a few times. I liked the city very much...in recent years, it has become more and more cosmopolitan...the downtown area is very hep...
For apartments, you can check out
Find Apartments For Rent, Rentals, Roommates, Moving Companies and More - ForRent.com. I'm sure you'll get tons of places in there. When I was apartment hunting here, that used to be a major source where I would look.
Since your kid is 3 yrs old, he can join a preK (pre kindergarten) class near by. They have several activities to engage kids.
You can also become a member of a local library (its free btw). Check with them if they have any activities for 3+ kids.
Hope this helps.
Aarushi