So I got my first drone and was checking out the area where I live. It looks like my house and a good part of the surrounding area is in a class D zone. If I read correctly, I need to contact ATC for the municipal airport in my town to fly in this zone. Is this correct?

I live in the enhanced warning zone in the picture.

Category D Airspace

The most common controlled airspace zone that will affect where a pilot is allowed to fly their UAV is category D.  If the flight involves this airspace, then the pilot must apply ahead of time for special permissions to fly if their drone is in the 7-20 kg category.  If your drone pilot is flying a drone that weighs under 7kg then these permissions aren’t necessary (the rules governing safe flight at all times still apply of course).  Your pilot will be able to advise you the crafts they have in their fleet and what they would plan on using.