Stepcounter after paragraph

\paragraph (like most headings) does not actually start a paragraph by setting anything. This is done to avoid headings ending up at the bottom of the page and being disconnected from what follows it. You'll have to set some regular text (even if that text is an empty box \mbox{}, say) in order for the paragraph headings to actually be set, and therefore display the correct numbering sequence.

\documentclass{article}

\newcounter{mycounter}
\setcounter{mycounter}{1}

\begin{document}

\paragraph{\themycounter}
Something% You need something here to set the above \paragraph heading

\stepcounter{mycounter}%
\paragraph{\themycounter}

\end{document}

\paragraph, like all sectioning commands, already has a dedicated counter that it increments so all you need is:

\documentclass{article}

\setcounter{secnumdepth}{4}
\renewcommand\theparagraph{\arabic{paragraph}}

\begin{document}
    \paragraph{}

    \paragraph{}
\end{document}

However as \paragraph is a 4th level heading it should not be used, except after the 3rd level heading, \subsubsection