Use \Q to autoescape any potentially problematic characters in your variable.
if($text_to_search =~ m/\Q$search_string/) print "wee";
Update: To clarify how this works…
The \Q will turn on “autoescaping” of special characters in the regex. That means that any characters which would otherwise have a special meaning inside the match operator (for example, *, ^ or [ and ]) will have a \ inserted before them so their special meaning is switched off.
The autoescaping is in effect until one of two situations occurs. Either a \E is found in the string or the end of the string is reached.
In my example above, there was no need to turn off the autoescaping, so I omitted the \E. If you need to use regex metacharacters later in the regex, then you’ll need to use \E.