I think you've done a good job with your first Perl script! The comments above on your coding are excellent.

As a suggestion, always begin you scripts with the following:

use strict;
use warnings;
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I tend to prefer this, instead, which contains the above:

use Modern::Perl;
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In either case, these pragmas will catch problematic areas in your script, perhaps saving hours of debugging--among other things.

This being said, consider the following (which doesn't really do anything more than what's already been done above):

use Modern::Perl;

my $word = 'a';

my @myNames = qw(Jacob Michael Joshua Matthew Alexander Andrew);

if ( $word ~~ @myNames ) {
    # Using Perl's smart-matching operator
    say qq|Exact match of "$word" found.|;
}
elsif ( my @found = grep /\Q$word\E/i, @myNames ) {
    local $" = "\n"; # Printed between interpolated array's elements
    say qq|Substring/case-insensitive matches for "$word":\n@found|;
}
else {
    say qq|"$word" not found.|;
}
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Output:

Substring/case-insensitive matches for "a":
Jacob
Michael
Joshua
Matthew
Alexander
Andrew
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Actually, the primary difference in the above script is its using Perl's 'smart' operator to check if an element exists in an array.

Hope this helps!