How did we discover what an atom looks like?

We know somewhat about atoms, yet not it's exact shape.

We know that atoms have a nucleus: Geiger-Marsden experiment

The Oil drop experiment showed the charge of the electron.

The Proton was also found as part of the atom.

And in the early twentieth century there was the Discovery of the neutron

These components are confirmed by scientific research.

The actual looks of an atom is a very different story. This is something we don't know because the electron cannot be placed in a fixed place or a fixed orbit around the nucleus. And the nucleus is composed of various even smaller particles so we don't know the exact looks of the atom. A model, a simplification of reality, is often used to represent an atom.