Please read I know what I mean. Why don't you?
Then read Short, Self-Contained, Correct Example.

That said, if you retrieve xxx.cgi?action=one&reaction=two, the following

#!/usr/bin/perl

use CGI;

my $query = CGI->new;
my $action = $query->param('action');
my $reaction = $query->param('reaction');

print $query->header,
      $query->start_html,
      "<h1>Result</h1>\n",
      "Action: ", $action || "no action",
      "<br>",
      "Reaction: ", $reaction || "no reaction",
      $query->end_html;
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will - if the webserver is correctly configured, if xxx.cgi has the correct permissions etc - serve a page reading:

Result

Action: one
Reaction: two

Now let me comment the snippets you posted.

my GetLink = 'one&reaction=two';
query string xxx.cgi?action=' + GetLink;
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doesn't compile. Running that yields:

No such class GetLink at - line 1, near "my GetLink"
syntax error at - line 1, near "my GetLink ="
Can't find string terminator "'" anywhere before EOF at - line 2.
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Then,

my $query = new CGI;
my $action = lc ($query->param('action'));

.......

elsif ($action eq "one"){
  my ($string1, $string1 ) = @_;
  my ($string2, $string2 ) = @_;
  warn("string2 = '$string2'");
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declaring a variable twice as in my ($string1,$string1) makes no sense at all. The special array @_ should only be used inside a subroutine and contains its arguments, if any. See perlvar and perlsub. Actually, don't see, read those pages.

Lastly,

elsif ($action eq "one"){
  my $query = CGI->new();
  my $string2 = $query->param('reaction');
  warn("string2 = '$string2'");
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here you are masking the outer $query with a new CGI object inside the elsif block. Why?