See, if the following solution is usuable for you:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{caption}
\usepackage{array,makecell}
\renewcommand\theadfont{\bfseries\normalsize}
\begin{document}
\begin{table}
\setcellgapes{5pt}
\makegapedcells
\caption{Pronombres personales.}
\label{tab:personales}
\centering
\begin{tabular}{|>{\centering\arraybackslash}p{2cm}
|>{\centering\arraybackslash}p{4cm}
|>{\centering\arraybackslash}p{5cm}|
}
\hline
\multicolumn{3}{|c|}{\thead{Pronombres}} \\ \hline
\thead{Inglés} & \thead{Pronunciación} & \thead{Español} \\ \hline
I & ái: & Yo \\ \hline
He & ji & Él \\ \hline
She & shi & Ella \\ \hline
We & wi & Nosotros(as) \\ \hline
You & iu & Tú, usted(es),
vosotros(as) \\ \hline
They & de$\underline{i}$ & Ellos o Ellas \\
\hline
\end{tabular}
\end{table}
\end{document}
Edit:
Problem of your MWE is mess of inproiper use \vspye*{2mm}. Similar solution as above you can also achieve by use \renewcommand\arraystretch{1.5}:
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{caption}
\usepackage{array,makecell}
\renewcommand\theadfont{\bfseries\normalsize}
\begin{document}
\begin{table}
\renewcommand\arraystretch{1.5}
\caption{Pronombres personales.}
\label{tab:personales}
\centering
\begin{tabular}{|>{\centering\arraybackslash}m{2cm}
|>{\centering\arraybackslash}m{4cm}
|>{\centering\arraybackslash}m{5cm}|
}
\hline
\multicolumn{3}{|c|}{\thead{Pronombres}} \\ \hline
\thead{Inglés} & \thead{Pronunciación} & \thead{Español} \\ \hline
I & ái: & Yo \\ \hline
They & de$\underline{i}$ & Ellos o Ellas \\
\hline
\end{tabular}
\end{table}
\end{document}
as is mentioned in a @thewaywewalk comment. With this you will avoid eventual problems with incompability between macros \setcellgapes{5pt} \makegapedcells and m column type. Result is the same as before.
vspacecommands? you should have a look into\renewcommand{\arraystretch}{<value>}– Robert Seifert Jul 17 '16 at 09:33