(Abridged) We use new multi-wavelength radio observations, made with the VLA and Effelsberg telescopes, to study the magnetic field of the nearby galaxy M51 on scales from $200pc$ to several $kpc$. Interferometric and single dish data are combined to obtain new maps at wwav{3}{6} in total and polarized emission, and earlier wav{20} data are re-reduced. We compare the spatial distribution of the radio emission with observations of the neutral gas, derive radio spectral index and Faraday depolarization maps, and model the large-scale variation in Faraday rotation in order to deduce the structure of the regular magnetic field. We find that the wav{20} emission from the disc is severely depolarized and that a dominating fraction of the observed polarized emission at wav{6} must be due to anisotropic small-scale magnetic fields. Taking this into account, we derive two components for the regular magnetic field in this galaxy: the disc is dominated by a combination of azimuthal modes, $m=0+2$, but in the halo only an $m=1$ mode is required to fit the observations. We disuss how the observed arm-interarm contrast in radio intensities can be reconciled with evidence for strong gas compression in the spiral shocks. The average arm--interam contrast, representative of the radii $r>2kpc$ where the spiral arms are broader, is not compatible with straightforward compression: lower arm--interarm contrasts than expected may be due to resolution effects and emph{decompression} of the magnetic field as it leaves the arms. We suggest a simple method to estimate the turbulent scale in the magneto-ionic medium from the dependence of the standard deviation of the observed Faraday rotation measure on resolution. We thus obtain an estimate of $50pc$ for the size of the turbulent eddies.
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