My math inline text is smaller than the normal text!

The x-height of the sans-serif font you're working with -- tgheros, an Helvetica clone -- is much larger than the x-height of your math font, which is Computer Modern. As far as I know, the tgheros package doesn't provide a scaling option. However, if you replace

\usepackage{tgheros}

with

\usepackage[scaled=0.78]{helvet}

and change the document font size from 11pt to 14pt, the scaling mismatch will have been alleviated significantly. (As you can probably guess, the helvet package provides another Helvetica clone.)

For my opinion this font Antykwa Półtawskiego for the text is better than helvetica.

   \documentclass[a4paper,11pt]{extarticle}
    \usepackage[utf8]{inputenc}
    \usepackage[T1]{fontenc}
    \usepackage{graphicx}
    \usepackage{xcolor}
    \usepackage{tikz}
    \usepackage{pgf}
    \usepackage[left=2cm,top=1cm,right=2cm,nohead,nofoot]{geometry}
    \usetikzlibrary{arrows,tikzmark}
    \usetikzlibrary{calc,trees,positioning,arrows,chains,shapes.geometric,%
        decorations.pathreplacing,decorations.pathmorphing,shapes,%
        matrix,shapes.symbols}
    \usepackage{multirow}
    \usepackage{amsmath,amssymb,textcomp}
    \everymath{\displaystyle}

    \usepackage{times}
    \renewcommand{\familydefault}{\sfdefault}
    \usepackage{antpolt}
    \usepackage[defaultmono,scale=0.85]{droidmono}

    \usepackage{multicol}
    \setlength{\columnseprule}{0pt}
    \setlength{\columnsep}{20.0pt}
    \usepackage{geometry}
    \geometry{
    a4paper,
    total={210mm,297mm},
    left=10mm,right=10mm,top=10mm,bottom=15mm}

    \linespread{1.3}
    \usepackage{tcolorbox}

\begin{document}

Vi har altså at både $a$ og $b$ er positive tal og $a$ er forskellig fra $1$.\\
Konstanten $a$ kaldes for \textcolor{blue!75!black}{\textbf{fremskrivningsfaktoren}}(eller grundtallet), mens konstanten $b$ kaldes for \textcolor{blue!75!black}{\textbf{begyndelsesværdien}} fordi grafen for den ekspoenetielle funktion skærer y-aksen i punktet $(0,b)$, dette kan vi vise på følgende måde ved at indsætte $x=0$ i forskriften.
\begin{center}
$f(0)=b\cdot a^0=b\cdot 1=b$
\end{center}
\end{document}