How to write after a figure in latex?

As @leandriis has already pointed out in a comment, removing the vspace directives, providing the [htb] placement specifier for the figure environment, and placing the tabular environment next to instead of below the tikzpicture environment are all worth doing. In addition, you may also want to assign a table-caption rather than a figure-caption to the tabular material.

documentclass{article}
usepackage{pgfplots,caption}
begin{document}

section{exponents}
bla bla bla 

begin{figure}[htb!]
begin{minipage}{0.5textwidth}
begin{tikzpicture}
    begin{axis}[name=plot,
    xmin=0, xmax=10,
    ymin=0,ymax=1000,width=0.9textwidth,
    legend pos=north west,]
    addplot [domain=0:10, color=blue, thick]   {5*x};
    addlegendentry{5x}
    addplot [domain=0:10, samples=100, color=red, thick]   {x^2};
     addlegendentry{$x^2$}
    addplot [domain=0:10, samples=100, color=black, thick]   {2^x};
     addlegendentry{$2^x$}
    end{axis}
    end{tikzpicture}
caption{aaaa label{fig:plot1}}
end{minipage}%
begin{minipage}{0.5textwidth}
centering
begin{tabular}{ |c|c|c|c| } 
    hline
    n & 5x & $x^2$ & $2^n$   
    hline
    3 & 15 & 9 & 8   
    5 & 25 & 25 & 32   
    7 & 35 & 49 & 128   
    9 & 45 & 81 & 512   
    11 & 55 & 121 & 2048   
    13 & 65 & 169 & 8192   
    15 & 75 & 225 & 32768   
    hline
end{tabular}
captionof{table}{bbbb label{fig:plot2}}
end{minipage}
end{figure}

section{polynomials}

more bla bla bla

end{document}