It doesn't appear that you need to redirect STDOUT in this case. You can open a file handle, and then print directly to it:

use warnings;
use strict;

open my $wfh, ">>", "output_cal_date.txt" or die "$0: open : $!";
my $currenttime = localtime();

print $wfh "Current Time : $currenttime\n\n"; # prints to file

print "this prints to STDOUT";

print ("Enter the FROM date in dd/mm/yyyy format : \n");
my $from_date_1 = <STDIN>;
chomp($from_date_1);

print $wfh $from_date_1;

close $wfh;
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