The European x-ray free-electron laser (XFEL) provides x-ray self-amplified spontaneous emission (SASE) FEL radiation in the wavelength range from 0.1 to 3 nm using three undulator systems. The SASE mode of operation at the European XFEL defines specific behavior of longitudinal and transverse coherence properties. In this paper, we describe the evolution of the temporal and transverse correlation functions along the undulator length, and we extract the corresponding evolution of coherence time and degree of transverse coherence as typical figures of merit. Generation of coherent radiation inside the FEL undulators is followed by beam transport to the experiments. During transport, the total number of coherent modes is preserved, but the wavefront can be disturbed, and we analyze the conditions under which this occurs. It is emphasized that the development of experimental observables for the degree of coherence and wavefront properties will be important for experiments using coherent x-ray radiation.