Other people have pointed you to File::Spec->catfile: that's the way to do it for portable code (e.g. something you expect to post to CPAN).

What I do for local, unix-only hackery is: I never include a trailing slash on directories, and always build up new paths with double-quoted strings.

A typical script might look something like this:

use Env qw( $HOME );
use File::Path qw( mkpath );
use File::Basename qw( basename dirname fileparse );
use File::Copy qw( move copy );

my $some_file = shift;
my $basename = basename( $some_file );
my $backup_location = "$HOME/backups";
mkpath( $backup_location ) unless -d $backup_location;

my $new_file = "$backup_location/$basename.bak";

copy("$some_file", "$new_file") or die "got problems: $!";
[download]

My personal opinion is that doing things like this is a win for readability:
my $new_file = "$backup_location/$basename.bak";

Also, note the stack of "use"s at the top... it's hard to do anything The Right Way in perl without leaning on the standard library a lot. These days I use script templates that include a dozen "use"s by default, so I don't have to try to remember what module "fileparse" is hidden inside of, and so on.