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Let $Sch_{Zar}, Sch_{et}$ denote scheme with Zariski and Etale topology respectively. Is there a functor from $Sch_{et}$ to $Sch_{Zar}$ (or from $Sch_{Zar}$ to $Sch_{et}$) which preserves fiber product? Furthermore, can we define a morphism of schemes from $X_{et}$ to $X_{Zar}$ where $X$ is a scheme? If so what are the properties of this morphism?

Any suggestions on reference where similar questions are studied will be most helpful.

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I don't think this is a very well posed question. For one, you probably don't mean a morphism of schemes from $X_\text{ét}$ to $X_\mathrm{Zar}$ but a morphism of sites.

In any case, an open immersion is étale, so there is a morphism from the Zariski site to the étale site. It obviously preserves fibered products. (Under mild hypotheses this is true for any morphism of sites.) All this means is that an étale sheaf may always be thought of as a Zariski sheaf.

Without knowing more details about what you are interested in it's hard to give a reference. But you could read:

  • Vistoli's chapter in FGA Explained
  • Mumford's 1965 paper "Picard groups of moduli problems"
  • Milne's online textbook on étale cohomology
  • Mac Lane and Moerdijk's "Sheaves in geometry and logic"
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