A great source on the history of particle physics is Pais's Inward Bound, see also references in In which experiments the charge to mass ratio of proton was determined? thread on Physics SE.
Charge to mass ratios for various ions were measured soon after Thomson's discovery of the electron, first for hydrogen apparently by Wien in 1898 based on canal (or anode) rays discovered by Goldstein in 1886. In 1902 Thomson measured it also for chlorine, nitrogen and neon, which led to the discovery of isotopes, see Atomic Structure. Of course, the ratios for ions depended on a gas, unlike the ones for electrons.
But the idea of a "fundamental" atom goes all the way back to 1815-16 when Prout, an English chemist, observed that the atomic weights were whole multiples of the atomic weight of hydrogen, and hypothesized that other elements were groupings of hydrogen atoms, which he called protyles (from Greek first matter). More careful measurements by Berzelius and Turner in 1828 and 1832 disproved the Prout hypothesis, but Rutherford was influenced by it. After the scattering experiments of 1911, when he conjectured existence of a nucleus surrounded by a large void, van den Broek proposed that the atomic number of an element is equal to its nuclear charge. This was experimentally confirmed by Moseley in 1913, and revived the Prout hypothesis, the Berzelius-Turner discrepancies could now be attributed to isotopes.
In 1917 experiments (reported in 1919) Rutherford was able to produce hydrogen nuclei by bombarding nitrogen with alpha-particles, thus confirming the Prout hypothesis directly. And in August 1920, at a meeting of the British Association for the Advancement of Science in Cardiff he modified Prout's "protyles" into "protons" (apparently, the play on "prouton" was involved) referring to hydrogen nuclei. The 1917 experiments are sometimes even billed as the "discovery" of proton. Chadwick discovered neutrons only in 1932, and the mystery of isotopes and deviations from Prout was finally solved.
Proton did not enjoy the "fundamental" status for very long, and not just because of quarks. For a long time after their first appearance in 1964 quarks were considered speculative entities, and their confinement made protons at least "unbreakable". But many grand unification schemes, starting with the $SU(5)$ one of Georgi-Glashow, predicted proton decay, and experimenters were chasing after it ever since.