If you want a subset of the data then either change the code immediately after the line parsing (it only saves things which are of the form p=v now). Or filter what you need at the final "print it" stage, which is what I did below in some more code for you. I decided to do it that way so that when problems happen latter, you have all the p=v pairs and you can use Dumper to see what is going on. There is no need here to optimize to reduce memory storage.

What to print is specified by an array (important_keys) and you can modify that as needed. If you want a special sort order, then modify the code I gave you earlier. BUT I would recommend against that in favor of a simple normal alphabetic sort. In my experience this usually works out best for "human readable" reports - humans scan sorted lists easily. Note: the hash important_order isn't used for a sort, but is used as an easy way to determine if some parm is in the "desired" list or not.

You've got some work to do, but I think you are on the right path.

As an update, to avoid sorting the final parms altogether, iterate through the important_keys array and print that key,value pair if it exists. The order in the array determines the "sort order" automatically (i.e. no sorting at all). Of course you could just sort the important_keys array in method 2 to get same as method 1, but you should know how to generically just print all the keys in this type of hash structure, so I left method 1 in.

#!/usr/bin/perl -w
use strict;
use Data::Dumper;

my %backups;

my @important_keys = qw(backup-status backup-set backup-date backup-ti
+me ERROR);

# this is how to translate the array into something that could
# be used for a special sort order, but I don't think you will
# need to do that.

my $i=1;
my %important_order = map{$_ => $i++} @important_keys;

while (<DATA>)
{
   next if (/^\s*$/);    #skip blank lines
   chomp;
   
   my ($date, $backupset , $parm , $value) = parseline($_);
   
   if ($value)
   {
      $backups{$backupset}{$parm} = $value;
   }   
}

#print Dumper \%backups;

foreach my $set (sort keys %backups)
{
    print "\n*** backup set: $set ***\n";
    
    #method 1
    foreach my $parm (sort keys %{$backups{$set}})
    {
       printf "  %-15s = %s \n", $parm, $backups{$set}{$parm}
          if $important_order{$parm};          
    }
     
     #method 2
  #  foreach my $parm (@important_keys)
  #  {
  #      printf "  %-15s = %s \n", $parm, $backups{$set}{$parm}
  #        if $backups{$set}{$parm};  
  #  }
}

sub parseline
{
   my $line = shift;
   my ($date, $rest) = $line =~ m/(^.*\d{4}):(.*)/;
   my ($backupset,  $msg) = split(/backup:INFO:/, $rest);
   $backupset =~ s/:\s*$//;  #trimming some unwanted thing like ':' is
+ ok
   $backupset =~ s/^\s*backup\.//; #more than one step is just fine to
+o!
   my ($parm, $value) = $msg =~ m/\s*(.*)=\s*(.*)\s*/;

   $parm  ||= $msg;   #if match doesn't happen these will be undef
   $value ||="";      #so this trick makes sure that they are defined.

   return ($date, $backupset, $parm, $value);
}   

=method 1 prints:
*** backup set: set1_lvm ***
  backup-date     = 20100816000003 
  backup-set      = backup.set1_lvm 
  backup-status   = Backup succeeded 
  backup-time     = 01:59:04 

*** backup set: set2_lvm ***
  backup-date     = 20100815200003 
  backup-set      = backup.set2_lvm 

*** backup set: set2_lvm_lvm ***
  backup-status   = Backup succeeded 
  backup-time     = 04:33:12 
  
=method 2 prints:
*** backup set: set1_lvm ***
  backup-status   = Backup succeeded 
  backup-set      = backup.set1_lvm 
  backup-date     = 20100816000003 
  backup-time     = 01:59:04 

*** backup set: set2_lvm ***
  backup-set      = backup.set2_lvm 
  backup-date     = 20100815200003 

*** backup set: set2_lvm_lvm ***
  backup-status   = Backup succeeded 
  backup-time     = 04:33:12 

=cut
[download]

__DATA__
Sun Aug 15 20:00:03 2010: backup.set2_lvm:backup:INFO: START OF BACKUP
Sun Aug 15 20:00:04 2010: backup.set2_lvm:backup:INFO: backup-set=back
+up.set2_lvm
Sun Aug 15 20:00:04 2010: backup.set2_lvm:backup:INFO: backup-date=201
+00815200003
Sun Aug 15 20:00:04 2010: backup.set2_lvm:backup:INFO: backup-type=reg
+ular
Sun Aug 15 20:00:04 2010: backup.set2_lvm:backup:INFO: backup-date-epo
+ch=1281927603
Sun Aug 15 20:00:04 2010: backup.set2_lvm:backup:INFO: backup-director
+y=/home/backups/backup.set2_lvm/20100815200003
Mon Aug 16 00:00:04 2010: backup.set1_lvm:backup:INFO: START OF BACKUP
Mon Aug 16 00:00:05 2010: backup.set1_lvm:backup:INFO: backup-set=back
+up.set1_lvm
Mon Aug 16 00:00:05 2010: backup.set1_lvm:backup:INFO: backup-date=201
+00816000003
Mon Aug 16 00:00:05 2010: backup.set1_lvm:backup:INFO: backup-type=reg
+ular
Mon Aug 16 00:00:05 2010: backup.set1_lvm:backup:INFO: backup-date-epo
+ch=1281942003
Mon Aug 16 00:33:15 2010: backup.set2_lvm_lvm:backup:INFO: last-backup
+=/home/backups/backup.set2_lvm_lvm/20100814200003
Mon Aug 16 00:33:15 2010: backup.set2_lvm_lvm:backup:INFO: backup-size
+=424.53 GB
Mon Aug 16 00:33:15 2010: backup.set2_lvm_lvm:backup:INFO: backup-time
+=04:33:12
Mon Aug 16 00:33:15 2010: backup.set2_lvm_lvm:backup:INFO: backup-stat
+us=Backup succeeded
Mon Aug 16 00:33:15 2010: backup.set2_lvm_lvm:backup:INFO: Backup succ
+eeded
Mon Aug 16 00:33:16 2010: backup.set2_lvm_lvm:backup:INFO: END OF BACK
+UP
Mon Aug 16 01:59:07 2010: backup.set1_lvm:backup:INFO: last-backup=/ho
+me/backups/backup.set1_lvm/20100815000006
Mon Aug 16 01:59:07 2010: backup.set1_lvm:backup:INFO: backup-size=187
+.24 GB
Mon Aug 16 01:59:07 2010: backup.set1_lvm:backup:INFO: backup-time=01:
+59:04
Mon Aug 16 01:59:07 2010: backup.set1_lvm:backup:INFO: backup-status=B
+ackup succeeded
Mon Aug 16 01:59:07 2010: backup.set1_lvm:backup:INFO: Backup succeede
+d
Mon Aug 16 01:59:09 2010: backup.set1_lvm:backup:INFO: END OF BACKUP
[download]