I'll let others focus on bench marking and foibles of the regex engine. I note though that you do more work than needed. Consider:

use warnings;
use strict;

my $fileText = <<TEXT;
Dear monks,

I'm trying to search for several regexes in some long files. To speed 
+things up,
I tried first checking a combined regex, to see if any of them matches
+ the line.
This seems to speed things up, at least when the regexes are just plai
+n words.
But: If I try input regexes which are anchored ("^word"), then suddenl
+y it's
much slower! Is there some weirdness with alternations and anchors? Or
+ did I
make some obvious mistake?
(I could rewrite ^aaa|^bbb|^ccc as ^(aaa|bbb|ccc), but it might be tha
+t only
some of the inputs are anchored.) Like so:
TEXT

my @argList = qw(regexes that ^I);
my $comb = join ('|', @argList);

print ">>> OP version\n";
open my $infile, "<", \$fileText;

while (my $line = <$infile>) {
  next if $line !~ $comb;

  for my $target(@argList) {
    next if $line !~ $target;
        print "Matched '$target' in: $line";
  }
}

print "\n>>> Grandfather version\n";
open $infile, "<", \$fileText;

while (my $line = <$infile>) {
  print "Matched '$1' in: $line" while $line =~ /($comb)/g;
}
[download]

Prints:

>>> OP version
Matched 'regexes' in: I'm trying to search for several regexes in some
+ long files. To speed things up,
Matched '^I' in: I'm trying to search for several regexes in some long
+ files. To speed things up,
Matched '^I' in: I tried first checking a combined regex, to see if an
+y of them matches the line.
Matched 'regexes' in: This seems to speed things up, at least when the
+ regexes are just plain words.
Matched 'regexes' in: But: If I try input regexes which are anchored (
+"^word"), then suddenly it's
Matched 'that' in: (I could rewrite ^aaa|^bbb|^ccc as ^(aaa|bbb|ccc), 
+but it might be that only

>>> Grandfather version
Matched 'I' in: I'm trying to search for several regexes in some long 
+files. To speed things up,
Matched 'regexes' in: I'm trying to search for several regexes in some
+ long files. To speed things up,
Matched 'I' in: I tried first checking a combined regex, to see if any
+ of them matches the line.
Matched 'regexes' in: This seems to speed things up, at least when the
+ regexes are just plain words.
Matched 'regexes' in: But: If I try input regexes which are anchored (
+"^word"), then suddenly it's
Matched 'that' in: (I could rewrite ^aaa|^bbb|^ccc as ^(aaa|bbb|ccc), 
+but it might be that only
[download]