Both are correct and they both mean you're going to the doctor's office.

@chinesespeakerhy's comment above is correct. Both are possible. 'Go to the doctor' refers to the person himself/herself, whereas 'go to the doctor's' refers to the doctor's home, clinic, surgery or consulting rooms (ie the place), but the meaning is the same.

The second sentence is correct but the first one isn't. The second sentence is not correct, its pronunciation will sound like you meant many doctors but not one.

Both are grammatically right. With 'Go to the doctor's', you are going to his work place. With 'go to the doctor', you are finding a doctor to diagnose you and treat you.