The growth by molecular beam epitaxy of GaAs on Si(100) has been studied by in situ scanning electron microscopy. The island growth process and subsequent surface roughness evolution was observed for substrate temperatures ranging from 375 \ifmmode^\circ\else\textdegree\fi{}C to 500 \ifmmode^\circ\else\textdegree\fi{}C. Two types of islands are formed: those that cease to grow beyond about 20 nm in size, and those that continue to grow, eventually forming a continuous layer. The large islands show preferred facets and grow slowly unless they merge with neighboring islands. After merging, they grow quickly until they complete the preferred facet again. Continuous layers are formed with small-scale roughness originating from the outline of the large islands. This evolution is quantified by calculating autocorrelation lengths. This characteristic length scale of the roughness grows with time, showing little variation with substrate temperature at early times. The in-plane roughness is seen to scale as a power law in time with exponent $1/z=0.29\ifmmode\pm\else\textpm\fi{}0.01.$ Like the original islands, the roughness scale grows by the successive merger of facetted structures with neighbors. The roughness evolution is compared qualitatively and quantitatively to self-affine models and mound coarsening models.