Evidence for the standard model comes mainly from particle physics experiments. Deep inelastic scattering experiments with electrons conducted in the 1960s implied that protons, neutrons and mesons contained internal structure. Because the size of these particles is is very small, very short wavelength/high energy electrons were used.
These experiments measured the probability that a given constituent of the proton carries a fraction of the proton's total momentum . The results were consistent with an internal structure consisting of three quarks, each with one third of the momentum of the proton.
Observation on the electromagnetic interaction revealed a weaker interaction than if each constituent part of the neutron/proton/meson carried a unit charge, equal in magnitude to the electron charge. Investigations implied that the hypothesized quarks carried charge in multiples of one third of the electron charge.
Electrons penetrating the particles were deflected at very small angles, indicating that the quarks were rebounding and could move freely while inside the nucleus. The closer together the quarks, the more they could rebound.
Evidence for quark colour came from looking at the probability of an electron scattering off a nucleus. The probability was larger by a factor of three than expected. This was explained by hypothesising that quarks came in three colours, each able of making a contribution to the probability of scattering an electron.
The interaction between the electron the the quarks inside the nucleus is electromagnetic and the size of this interaction could be measured and the momentum of the interacting particles deduced. For the proton and neutron this was less than the actual momentum of the proton/neutron, implying other particles within the nucleus. This was taken to be evidence of gluons, transmitting the strong force between the quarks.
Very strong evidence came from observations of 'neutral currents', produced by the Z0 particle, which could only properly be explained by the standard model.
The Standard Model though is still at best an incomplete theory. It cannot predict or explain the properties of particles – mass and charge - from first principles. These always have to be measured by experiment. Other theories may not have this flaw, and may then be preferred to the Standard Model.