You could use Email::MIME, but I think you'll somehow have to handle the separate cases...  As multipart messages can in theory be arbitrarily nested, a recursive approach is appropriate.  The following example returns the first acceptable (i.e. plain text or html) part found:

use Email::MIME;

sub handle_parts {
    my $part = shift;
    my $content_type = $part->content_type;
    #print "Content-Type: $content_type\n";  # debug

    my $body = $part->body;
    
    if ($content_type =~ m#text/plain#) {
        return $body;
        
    } elsif ($content_type =~ m#text/html#) {
        return html2text($body);
        
    } elsif ($content_type =~ m#multipart/#) {
        for my $subpart ($part->parts) {
            my $text = handle_parts($subpart);
            return $text if defined $text;
        }
    }
    return;
}

sub html2text {
    my $html = shift;
    # my $text = ...  (left as an exercise)
    # return $text;
    return $html;
}

my $message = ...

my $parsed = Email::MIME->new($message);

my $text = handle_parts($parsed);
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You might need some additional checks to figure out if the first content part is the one you want...