Would you mind weaving it into my code so I can see for myself?

(Note: The following provides answers already given in essence by LanX, but it at least also provides an example of sorts and so may be helpful.)

... how would I use [data dumper] to see the data and the structure of $ua and $res?

Untested:

use Data::Dumper;
...
my $ua = LWP::UserAgent->new;
print Dumper $ua;
...
my $res = $ua->get(...);
print Dumper $res;
...
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Also a point of confusion: is
    $ua->show_progress
the same thing as
    $$ua{show_progress}
?

It is not. The expression  $ua->show_progress invokes (without any arguments) the method  show_progress() in the class (or a superclass) of which  $ua is an object reference.

The expression  $$ua{show_progress} attempts to access the value of the  'show_progress' key in the anonymous hash to which  $ua is presumed to be a reference; it is an error if  $ua is not a hash reference. This expression is the same as the form  $ua->{show_progress} which is preferred (properly, IMHO).

c:\@Work\Perl>perl -wMstrict -le
"{ package Foo;
   use Data::Dump qw(pp);
   ;;
   sub new {
     my $class = shift;
     return bless { show_progress => 'not so fast',  @_, } => $class;
     }
   sub show_progress {
     my $self = shift;
     print 'Real Thing: ', pp $self;
     }
   }
 ;;
 my $object_reference = Foo->new(foo => 'bar',  hi => 'lo');
 ;;
 $object_reference->show_progress;
 print 'A: ', $$object_reference  {show_progress};
 print 'B: ',  $object_reference->{show_progress};
"
Real Thing: bless({ foo => "bar", hi => "lo", show_progress => "not so
+ fast" }, "Foo")
A: not so fast
B: not so fast
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I see from Need help with loop syntax errors, that you like to experiment. I suggest that you do more of the same with Data::Dumper to empirically see what it does with various data structures.

In general you give Dumper a reference to any arbitrary structure and it figures out how to display it.

Here is some "play" code to get you started:

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

my @data = qw( 7 8 9);
my $dataref = \@data;      #create reference to data array
my $string = 'this is string';

print Dumper \@data;  #dump using reference to @data

=PRINTS
$VAR1 = [
          '7',
          '8',
          '9'
        ];
=cut

print Dumper $dataref; #dump using scalar reference

=PRINTS
$VAR1 = [
          '7',
          '8',
          '9'
        ];
=cut

print Dumper @data;

=PRINTS
$VAR1 = '7';
$VAR2 = '8';
$VAR3 = '9';
=cut

print Dumper $string;

=PRINTS
$VAR1 = 'this is string';
=cut

print Dumper \$string;

=PRINTS
$VAR1 = \'this is string';
=cut

__END__
I used a feature of Perl called perlpod - the "Plain Old Documentation
+"
commands to interleave multi-line printouts into the code. That is not
what this is normally used for, but you will sometimes see this techni
+que
on PM as a way to make the code and the printout all "one Perl file" t
+hat
can be executed.
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