Maybe there is package somewhere that does this kind of stuff out of the box.
Update: There is -- the bibunits package would let you do this in the same document, but still require multiple runs of bibtex.
However, if there isn't, the easiest way may be to cheat a bit and create a helper document that just contains the selected references, and include the bibliography produced by that document in the main text. Example:
Helper document dummy.tex:
\documentclass{article}
\usepackage{lipsum}
\begin{document}
\bibliographystyle{plain}
\nocite{Palmer2013,Zhang2014} % name all your selected references
\bibliography{references}
\end{document}
Running latex and bibtex on dummy.tex will give you the selected reading list inside dummy.bbl.
Main document:
\documentclass{article}
\usepackage{lipsum}
\begin{document}
\bibliographystyle{plain}
\section{Introduction}
\lipsum[1]
We discuss a couple of papers in the main text \cite{Palmer2013,Zhang2014,Taylor2016,Zitzer2001}.
\lipsum[2]
\clearpage
\bibliography{references}
% change the title for included reading list
\renewcommand{\refname}{Selected reading} % for documentclass article
%\renewcommand{\bibname}{Selected reading} % for documentclass book or report
% dummy.bbl is generated by running latex and bibtex on dummy.tex
\input{dummy.bbl}
\end{document}
