I want to know what are alternatives for this task and what would you do differently (better).

That's a pretty wide open question the answer to which depends on a lot of factors. More factors than I'd care to enumerate here.

One of my personal favorites for working with files (although I use opendir and plain old open more often is Tie::File which allows you to treat a file as if it were an array. Pretty slick if you ask me.

That said I am an adherent to the old adage "right tool for the job." If open does the job, and it does really well, then use it. Don't get fancy unless you have a real need to.

There is a real temptation when you spot something "neat" that you end up having a solution in search of a problem. A good example of that is when I got my first router and router table for wood working. Pretty soon I was doing all sorts of fancy edging work on every wood project I had. I realized I'd overstepped the utility of it when I used a coving bit to round off the edges really pretty for something that was going to be out of sight anyway.

Writing code is like that too. We get tempted to use that fancy programming technique when brute force is a quicker way to get the job done.

Just some thoughts...

Generally, your code is OK, but I would like to point out some things which can be improved.

#declare the filehandeles outside the foreach loop

opendir (DIR, $dir) or die $!;
my @textFiles = grep /\.txt/, readdir DIR;
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By the way, you can open just "." instead of using Cwd. "." means "current directory" in modern operating systems.

    my $text_file_out = $text_file;
    $text_file_out =~ s/\.txt//;
    $text_file_out = $text_file_out . '_new.txt';
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what would you do differently

#!/usr/bin/perl
use warnings;
use strict;

#test file to learn how to process multiple files at once

#first get the path for current directory
my $dir = '.'; # current working directory

#open and read only the .txt files
opendir my $DIR, $dir or die "Cannot open '$dir' because: $!";
my @textFiles = grep /\.txt\z/, readdir $DIR;
closedir $DIR or die "Could not close '$dir' because: $!";

#open each file for further processing
foreach my $text_file ( @textFiles ) {
    open my $text_file_fh_in, '<', $text_file or die "Cannot open '$te
+xt_file' because: $!";  #open for reading

    #create the names of the new files
    my $text_file_out = $text_file;
    $text_file_out =~ s/(?=\.txt\z)/_new/;

    open my $text_file_fh_out, '>', $text_file_out or die "Cannot open
+ '$text_file_out' because: $!";  #open for writing

    #tests to see if it works
    #print "$text_file\n"; 
    print "$text_file_fh_in\n";
    #print "$text_file_out\n"; 
    print "$text_file_fh_out\n";

    while ( <$text_file_fh_in> ) {
        if ( /^(.*?)\.(.*?)\ (.*?)\.(.*?)\ (.*?)$/ ) {
            print $text_file_fh_out "$1\t$2\t$4\t$5\n";
        } 
    }

    #close the filehandles
    close $text_file_fh_in or die "Could not close '$text_file' becaus
+e: $!";  
    close $text_file_fh_out or die "Could not close '$text_file_out' b
+ecause: $!";
}
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I'd probably do it something like this. (Untested code...)

use strict;
use warnings;
use aliased 'Path::Class::Rule';
use Path::Class qw( dir file );

my $files = Rule->new->file->name(qr{\.txt$})->iter;
while ( my $file = $files->() )
{
    (my $newfile = $file) =~ s/\.txt$/_new.txt/;
    $newfile = file($newfile);
    
    print "$file  ->  $newfile\n";
    
    $newfile->spew(
        map {
            s/^(.*?)\.(.*?)\ (.*?)\.(.*?)\ (.*?)$/$1\t$2\t$4\t$5/; $_
        } $file->slurp;
    );
}
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