Cancer treatment frequently fails due to the evolution of drug-resistant cell phenotypes driven by genetic or non-genetic changes. The origin, timing, and rate of spread of these adaptations are critical for understanding drug resistance mechanisms but remain challenging to observe directly. We present a mathematical framework to infer drug resistance dynamics from genetic lineage tracing and population size data without direct measurement of resistance phenotypes. Simulation experiments demonstrate that the framework accurately recovers ground-truth evolutionary dynamics. Experimental evolution to 5-Fu chemotherapy in colorectal cancer cell lines SW620 and HCT116 validates the framework. In SW620 cells, a stable pre-existing resistant subpopulation was inferred, whereas in HCT116 cells, resistance emerged through phenotypic switching into a slow-growing resistant state with stochastic progression to full resistance. Functional assays, including scRNA-seq and scDNA-seq, validate these distinct evolutionary routes. This framework facilitates rapid characterisation of resistance mechanisms across diverse experimental settings.