documentclass[11pt]{amsart}
usepackage[paper=a4paper,left=35mm,right=25mm,top=40mm,bottom=40mm]{geometry}
begin{document}
tableofcontents
%
newpage
section{First Section}

subsection{First Subsection}

subsection{Second Subsection}

section{Second Section}

end{document}

documentclass[11pt]{amsart}
usepackage[paper=a4paper,left=35mm,right=25mm,top=40mm,bottom=40mm]{geometry}
begin{document}

tableofcontents
%
newpage
part{First Section}

section{First Subsection}

section{Second Subsection}

part{Second Section}

end{document}

documentclass[11pt]{amsart}
usepackage[paper=a4paper,left=35mm,right=25mm,top=40mm,bottom=40mm]{geometry}

makeatletter
defl@section{@tocline{1}{12pt plus2pt}{0pt}{}{bfseries}}% <- added

def@tocline#1#2#3#4#5#6#7{relax
    ifnum #1>-1% <- added
  ifnum #1>c@tocdepth % then omit
  else
    par addpenalty@secpenaltyaddvspace{#2}%
    begingroup hyphenpenalty@M
    @ifempty{#4}{%
      @tempdimacsname r@tocindentnumber#1endcsnamerelax
    }{%
      @tempdima#4relax
    }%
    parindentz@ leftskip#3relax advanceleftskip@tempdimarelax
    rightskip@pnumwidth plus4em parfillskip-@pnumwidth
    #5leavevmodehskip-@tempdima #6nobreakrelax
    hfilhbox to@pnumwidth{@tocpagenum{#7}}par
    nobreak
    endgroup
  fi% <- added
fi}

makeatother

begin{document}

tableofcontents
%
newpage
part{First Section}

section{First Subsection}

section{Second Subsection}

part{Second Section}

end{document}