An introduction to semi-Riemannian geometry and the general theory
of relativity can be found
here.
The introductory material (putting the Cauchy problem into
a historical context), as well as a sketch of the proof
of local existence is to be found in
The Cauchy problem in general relativity. Acta Phys. Polon. B44 (2013), no. 12, 2621--2641.
A preprint version is to be found here.
A historical description of the Cauchy problem can be found in the
article
"Origins and development of the Cauchy problem in general
relativity"
(pdf).
Class. Quantum Grav.32 (2015), 124003
Digital Object Identifier (DOI) 10.1088/0264-9381/32/12/124003.
For the journal version, see
here.
A sketch of a proof of future global non-linear stability in a cosmological
setting can be found in the article
"On proving future stability
of cosmological solutions with accelerated expansion" (pdf). Surveys in
Differential Geometry, Volume 20 (2015): One hundred years of general
relativity A jubilee volume on general relativity and mathematics.
Editors Lydia Bieri and Shing-Tung Yau.
For the journal version, see
here.
A more extensive sketch of a proof of local existence can be
found in Chapter 2 of the book
Ringström, Hans
On the Topology
and Future Stability of the Universe.
Oxford Mathematical Monographs. Oxford University Press, Oxford, 2013.
xiv+718 pp. ISBN: 978-0-19-968029-0
Moreover, that book contains a proof of local existence of solutions in
the Einstein-Vlasov setting.
For a proof of local existence of solutions in the vacuum setting, one reference
is
Ringström, Hans The Cauchy Problem
in General Relativity. ESI Lectures
in Mathematics and Physics. European Mathematical Society (EMS), Zürich,
2009. xiv+294 pp. ISBN: 978-3-03719-053-1
Concerning the existence of a maximal globally hyperbolic development, one
proof can be found here.
See also the recent proof of Jan
Sbierski which does not appeal to Zorn's lemma.