"There's a new Moose way of creating constructors, and defining what paramaters (arguments, options) those accept. Is there a Moose way to define subroutine parameters?"

No, but see MooseX::Method::Signatures, MooseX::Declare, etc.

"I've defined the attributes for my class... how do I deal with attributes of the object (i.e., self) that are objects from a different module?"

Maybe you want to be using lazy builders, and allowing for dependency injection...

package Example;
use Moose;
use Net::LDAP;

# Options related to SSH
has [qw( host user )] => (
   is       => "ro",
   isa      => "Str",
   required => 1,
);

has ldap_class => (
   is       => "ro",
   builder  => "_build_ldap_class",
);

has ldap => (
   is       => "ro",
   lazy     => 1,
   builder  => "_build_ldap",
);

has error => (
   is       => "rw",
   isa      => "Str",
);

sub _build_ldap_class {
   return "Net::LDAP";
}

sub _build_ldap {
   my $self = shift;
   return $self->ldap_class->new( $self->host );
}

sub get_user {
   my $self = shift;
   my $user = $self->user;
   my $host = $self->host;
   my $mesg = $self->ldap->bind;
   $mesg = $self->ldap->search(filter => "(uid=$user)");
   ...
}
[download]

Now if somebody using your class wants to change how it does LDAP searches, they can create a subclass of Net::LDAP and do:

my $eg = Example->new(ldap_class => "Net::MyLDAP", ...);
[download]

... and your Example class will magically use their subclassed Net::LDAP object.