We present a study of the low-frequency radio properties of star-forming (SF) galaxies and active galactic nuclei (AGNs) up to redshift z = 2.5. The new spectral window probed by the Low Frequency Array (LOFAR) allows us to reconstruct the radio continuum emission from 150 MHz to 1.4 GHz to an unprecedented depth for a radio-selected sample of 1542 galaxies in ~7deg2 of the LOFAR Bootes field. Using the extensive multiwavelength data set available in Bootes and detailed modelling of the far-infrared to ultraviolet spectral energy distribution (SED), we are able to separate the star formation (N = 758) and the AGN (N = 784) dominated populations. We study the shape of the radio SEDs and their evolution across cosmic time and find significant differences in the spectral curvature between the SF galaxy and AGN populations. While the radio spectra of SF galaxies exhibit a weak but statistically significant flattening, AGN SEDs show a clear trend to become steeper towards lower frequencies. No evolution of the spectral curvature as a function of redshift is found for SF galaxies or AGNs. We investigate the redshift evolution of the infrared-radio correlation for SF galaxies and find that the ratio of total infrared to 1.4-GHz radio luminosities decreases with increasing redshift: q1.4 GHz = (2.45 ± 0.04) (1 + z)-0.15 ± 0.03. Similarly, q150MHz shows a redshift evolution following q150 GHz = (1.72 ± 0.04) (1 + z)-0.22 ± 0.05. Calibration of the 150MHz radio luminosity as a star formation rate tracer suggests that a single power-law extrapolation from q1.4 GHz is not an accurate approximation at all redshifts.