  We construct moduli spaces of semistable objects on an Enriques surface for
generic Bridgeland stability condition and prove their projectivity. We further
generalize classical results about moduli spaces of semistable sheaves on an
Enriques surface to their Bridgeland counterparts. Using Bayer and Macr\`{i}'s
construction of a natural nef divisor varying with the stability condition, we
begin a systematic exploration of the relation between wall-crossing on the
Bridgeland stability manifold and the minimal model program for these moduli
spaces. We give three applications of our machinery to obtain new information
about the classical moduli spaces of Gieseker-stable sheaves:
  1) We obtain a region in the ample cone of the moduli space of
Gieseker-stable sheaves which works for all unnodal Enriques surfaces.
  2) We determine the nef cone of the Hilbert scheme of $n$ points on an
unnodal Enriques surface in terms of the classical geometry of its half-pencils
and the Cossec-Dolgachev $\phi$-function.
  3) We recover some classical results on linear systems on Enriques surfaces
and obtain some new ones about $n$-very ample line bundles.
