You have a great start on your script! As for your 'struggles': 1) yes, and ++ is a perfectly appropriate and a common construct; 2) The value for what? If you mean incrementing the hash value, you're doing it correctly. If you mean $row[18] to get the value of col 19, you're doing it correctly.; and 3) no.

Here are some suggested changes to consider for your script:

use strict;
use warnings;

my $filename = $ARGV[0];
my %gene_count;

open my $fh, '<', $filename or die "Cannot open $filename: $!";

while ( my $line = <$fh> ) {
    chomp;
    my @row = split( "\t", $line );
    $gene_count{ $row[18] }++ if $row[18];
}

close($fh);

print "$_ => $gene_count{$_}\n" for sort keys %gene_count;
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• $ARGV{0} -> $ARGV[0]
• Made a few changes to your open
• Added chomp because you're splitting on the tab character. If you don't chomp, a newline will be on the end of the array's last element (with the exception of the file's last line).
•  = ++ -> ++
• Added if $row[18] to check for 'good' key candidate. This check could be stronger, but is likely sufficient, in this case.
• Just fyi: The parens of split and close are optional.
• Added printing the sorted key/value pairs. (Just assumed you wanted to do that... :)

Since you're sending your script the filename from the command line, you can let Perl handle the file i/o. If you split on ' ' (whitespace) you don't need to chomp. Also, you can send split a LIMIT to its splitting, so it's not splitting all columns. Using this LIMIT can significantly speed the splitting process. Given this, the following is functionally equivalent:

use strict;
use warnings;

my %gene_count;

while (<>) {
    my @rows = split ' ', $_, 20;
    $gene_count{ $row[18] }++ if $row[18];
}

print "$_ => $gene_count{$_}\n" for sort keys %gene_count;
[download]

Your original script's logic is good; only minor fixes were needed. You've done well...

Hope this helps!