This questions is a bit hard to answer generally, because it all depends on the relative magnitudes of the relaxation times and the hyperfine-coupling (coupling between unpaired electrons and nuclei, analogous to spin-spin couplings in NMR). Also Ligand/solvent exchange plays a big role.
Roughly one can say that the lines of close nuclei will be broadened, sometimes so much that you won't see anything. If the relaxation of the electron spin is very fast, you will observe huge coupling constants, easily in the range of MHz, which means that you won't see the lines in normal NMR.
The influence on unbound ligands is not easily predicted. The main contribution probably comes from ligand exchange, but if your complex also binds solvent molecules, the peaks of the whole sample can be broadened. There are "paramagnetic relaxation enhancers" which are made exactly for that purpose in NMR, but mainly in MRI for contrast improvement.
for very distant spins, e.g. in proteins, the electron can lead to pseudo-contact shifts, which can be used as distant restraints.
EPR includes many techniques which give quite different information content. The Metalcenter and its surrounding can be characterized very well by CW-EPR, also the symmetry is quite easily extracted from the spectra (e.g. tetrahedral or quadratic-planar). for nuclei further away from the paramagnetic center see Babcock_Hall's post. There are pulsed EPR techniques which give a huge amount of structural information. In general these are not easily extracted and need simulations of the spectra.
All this can of course be understood very well in quantum mechanical framework.