First, people had to realize that the matter is composed of atoms. They had good reasons to think so for centuries. For example, the mixing ratios in chemistry were rational numbers (in some good enough units), indicating that a single material is made of small pieces of the same kind (atoms or molecules).
In the 19th century, the atomic theory of matter strengthened when it was shown that the statistical properties of the atoms and molecules may explain thermal phenomena. The energy per degree of freedom of a single atom is the temperature (times a numerical factor of order one and times Boltzmann's constant); the entropy is $k$ times the amount of information in "nats" (bits over the natural log of two).
In 1905 and 1906, the Brownian motion was explained as collisions of a pollen particle with the molecules of water, and the size of the molecules could have been estimated in this way, too. At that time, the serious opponents of the atomic theory became non-existent overnight.
The best microscopes today may see individual atoms directly. One just magnifies the view sufficiently (and uses high-frequency particles instead of visible photons so that the long waves don't make the picture fuzzy).
A few years later, Ernest Rutherford realized in his famous gold foil experiment (alpha radiation sometimes recoiled from gold foil in the opposite direction, proving that the gold must be made of very hard "localized" matter) that the atoms had a tiny positively charged nucleus, 10,000 times smaller than the atom, and it was orbited by (a) negatively charged particle(s), the electron(s). The nucleus was hypothesized to be made out of protons and neutrons. They were isolated by the 1932 discovery of the neutron.
In the late 1960s, deep inelastic experiments showed that much like atom has localized subparticles, protons and neutrons have localized much smaller particles inside, too. They were the partons or quarks. The quark-parton theory not only explained the deep inelastic scattering but also the classification of different hadrons (different composite particles similar to protons and neutrons; there are many of them). In some sense, this Gell-Mann's work on the "construction of hadrons out of quarks" was fully analogous to the atomic explanation of the Mendeleev periodic table of elements.
Different particles such as electron, its heavier cousins muon and tau, and the neutrinos, and different flavors of quarks etc. were discovered - and their masses were measured - at various moments of the history. The last known quark, the top quark, was discovered at the Tevatron in 1994. The last particle, the Higgs boson, was officially discovered on July 4th, 2012, by seeing bumps in the processes where a hypothetical new particle decays either to two photons or two Z bosons. In a large enough number of collisions, the LHC simply detects a pair of photons whose total center-of-mass energy is 126 GeV, thus proving that there must exist a new particle of this mass.
Neutrinos were harder to detect but they rarely interact with the nuclei which shows that they're present.
In some sense, your question is a very broad question asking "almost" about all of atomic and particle physics from the whole 20th century as well as big branches of thermodynamics etc. The details of the discovery of individual particles depend on the particle species. But a punch line is that it is not hard to "see" the elementary particles, almost directly. This point – seeing – is particularly explicit in the case of the microscopes seeing atoms; and in the case of charged elementary particles leaving tracks (of bubbles) in a cloud chamber etc. There are many ways how individual elementary particles manifest themselves.