You need to do at least three things to achieve the output you're after:
Make the page number reset with every use of \section. This can be achieved by
\numberwithin{page}{section}
The above also sets the page number display to be \thesection.\arabic{page} by default.
To correct the above default setting of the page numbering, use
\renewcommand{\thepage}{\thesection-\arabic{page}}
Finally, you want to let the page numbering start from 1. To do this, you need to make sure that the page numbering is adjusted only when you call \section (otherwise it may be reset when calling \subsection, \subsubsection...). Here a patch via etoolbox of \@sect inserts an appropriate page number stepping:
\makeatletter
\patchcmd{\@sect}% <cmd>
{\protected@edef}% <search>
{\def\arg{#1}\def\arg@{section}%
\ifx\arg\arg@\stepcounter{page}\fi%
\protected@edef}% <replace>
{}{}% <success><failure>
\makeatother
Here is complete, minimal example that shows the output:

\documentclass{article}
\usepackage[paper=a6paper]{geometry}% Just for this example
\usepackage{lipsum}% Just for this example
\usepackage{amsmath,etoolbox}
\numberwithin{page}{section}% Number page by section
\renewcommand{\thepage}{\thesection-\arabic{page}}% Page numbering style
\makeatletter
% Make sure that page starts from 1 with every \section
\patchcmd{\@sect}% <cmd>
{\protected@edef}% <search>
{\def\arg{#1}\def\arg@{section}%
\ifx\arg\arg@\stepcounter{page}\fi%
\protected@edef}% <replace>
{}{}% <success><failure>
\makeatother
\begin{document}
\section{A section}
\lipsum[1-5]
\section{Another section}
\lipsum[6-10]
\end{document}
\setcounter{page}{1}to force the counter to start at 1 instead of 0. I know how to alter the separator for Figures/Tables (e.g., seesubcaptionpackage documentation), so I imagine it's similar for page numbers. – honeste_vivere Oct 11 '14 at 19:01\thechapter,\thesection, and then\leftmarkand\rightmarkto figure this out. For details, you can check out the following link. Though, I should note that this question should be in the LaTeX forum, not Overflow. – honeste_vivere Oct 11 '14 at 19:07