Public transportation is mediocre, but people do get by without a car. Being along one of the main arteries (Westheimer or Main/Fannin for example), helps.
Best area and whether not having a car is doable all depends on where your job is relative to where you want to live. Some areas make it pretty easy to not have a car if you work fairly close by, especially in the age of Uber and if you are willing to sweat on a bike or by walking.
For example, if your job is in the Galleria area, you can pretty easily find an apartment in the area (albeit pricey) that allows you to forgo a car.
I suggest looking for identifying a job and its location and then looking to find an apartment nearby.