You should look into using the \tabcolsep length which determines the length between consecutive columns. That will produce that amount of space between two columns.
If however, you wish different spacing of different columns, you can use the p{<length>} specifier which sets the width of the column to <length>.
Below are examples of the normal table, with a \tabcolsep and with the p specifier.
\documentclass{article}
\begin{document}
\begin{tabular}{cc}
\hline
Hello & there\\\hline
where & am \\\hline
I & ? \\\hline
\end{tabular}
\qquad\tabcolsep=15pt
\begin{tabular}{cc}
\hline
Hello & there\\\hline
where & am \\\hline
I & ? \\\hline
\end{tabular}
\qquad
\begin{tabular}{p{1cm}p{2cm}}
\hline
Hello & there\\\hline
where & am \\\hline
I & ? \\\hline
\end{tabular}
\end{document}
Which will produce the following:

You should be able to use this for your symbol table.
If then you wish centering inside a p specifier you should use the array package and specify the columns as this (which will center the 1st column):
% \usepackage{array}
\begin{tabular}{>{\centering}p{1cm}p{2cm}}
\hline
Hello & there\\\hline
where & am \\\hline
I & ? \\\hline
\end{tabular}
The output will be:
