How did we discover what an atom looks like ?

There is no simple answer to your question. There is no "atomic microscope" which would allow one to look inside an atom and say, "Aha! There's 7 blue protons, 6 green neutrons, and 7 red electrons." The way the structure of the atom was devised was through a long series of experiments. Each one was designed to look at a specific aspect of the atom. At one time the atom was thought to be a solid ball of positive charge with electrons embedded in it. Then in 1909, Ernest Rutherford did an experiment which demonstrated that that picture was wrong and that the positive charge was centered at the center of the atom and occupied a very small volume compared to the whole atom.
Before the neutron was discovered in 1932, the nucleus was thought to have both protons and electrons in it. The number of protons was chosen to get the correct atomic weight and the number of electrons was chosen to get the correct nuclear charge. It turned out that this model did not give predictions that agreed with experiment. The discovery of the neutron lead to a revision of the model leading to the current one.

In the current model, the number of electrons in the atom is determined by gamma and x-ray spectroscopy. The number of protons in the atom is chosen to balance the charge of the electrons in the atom. The number of neutrons in the atom is chosen to give the correct atomic weight for the element in question. Many additional experiments were performed to confirm the model as finally developed and so far the results obtained are as one would expect from the model. This agreement between the experimental results and the predictions based on the model is what is called proof.

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.

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