You're very close. Here is one way to do it:

use strict;
use warnings;

my $string = 'This is "the search" string "that was" supplied';

my @quotes;

while( $string =~ s["(.*?)"][] )
{
    push( @quotes, $1 );
}

my @keys = split( ' ', $string );

use Data::Dumper;
print "Quotes:\n", Dumper( \@quotes );
print "Keys:\n", Dumper( \@keys );

output:

Quotes:
$VAR1 = [
          'the search',
          'that was'
        ];
Keys:
$VAR1 = [
          'This',
          'is',
          'string',
          'supplied'
        ];
[download]

I moved the '?' inside the parens so it modifies the '*'. Note also that s/// returns the number of substitutions made (see perlop), not the text that was substituted (or captured in $1, in this case), hence the while loop to walk through the string.

I'm sure someone else will come up with a more clever way of doing it, perhaps with one of the Text modules, but this works with the data you supplied. If you need to handle escaped quotes, definitely consider one of the modules.

HTH