Where do people first feel the ground shaking during an earthquake?

The strongest shaking is towards the center of the actual earthquake which it is called the focus.

The epicenter of an earthquake is the point on the Earth's surface above the focus or hypocenter which is the point within the Earth where the rupture occurs.

As such the epicenter is the closest point to this and the seismic waves have had to travel through the smallest amount of material and so have undergone the least amount of attenuation (in this case attenuation is a reduction in amplitude of the seismic waves due to energy being by moving through the Earth).

However as with a lot of things in Earth Sciences there can be exceptions to this! This is because the amplitude of seismic waves is affected by the density (related to the level of compaction in the case of soils) of the material through which it is traveling). So a seismic wave traveling through a hard rock such as granite will have a much lower amplitude than one traveling through a softer, less dense, poorly compacted material such as a soil and so due to the higher amplitude there will be a higher level of perceived and measurable ground shaking, even an equal distances from the epicenter.