Hello voidzero and welcome to the Monastery. Congratulations on beginning your Perl journey - many wonders await you!

Lots of the documentation is available in perldoc, both online and using the perldoc command in your terminal. The ^, & and ~ symbols in these cases are operators and you can read all about them in perlop.

Note that ^ is still a perfectly valid operator and not deprecated for general use. Our anonymous brother merely points out that the particular use case of $x ^ $x is not longer a valid method for determining the numerical nature of $x in the cases where $x is or may be a utf-8 string containing high code points. This is also true for the $x & ~$x method. For example this test:

use strict;
use warnings;

use Test::More tests => 6;

my $x = 111;
my $y = '111';
my $z = "1\N{LATIN CAPITAL LETTER GAMMA}1";

ok !($x & ~$x), '$x is a number, &~';
ok $y & ~$y, '$y is a string, &~';
ok $z & ~$z, '$z is a string, &~';

ok !($x ^ $x), '$x is a number, ^';
ok $y ^ $y, '$y is a string, ^';
ok $z ^ $z, '$z is a string, ^';
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runs perfectly well in perl 5.22 and earlier. However, starting at perl 5.24 it produces warnings like these:

$ perl isnum.t
1..6
ok 1 - $x is a number, &~
ok 2 - $y is a string, &~
Use of strings with code points over 0xFF as arguments to 1's compleme
+nt (~) operator is deprecated. This will be a fatal error in Perl 5.2
+8 at isnum.t line 12.
Use of code point 0xFFFFFFFFFFFFFFCE is deprecated; the permissible ma
+x is 0x7FFFFFFFFFFFFFFF. This will be fatal in Perl 5.28 at isnum.t l
+ine 12.
Use of code point 0xFFFFFFFFFFFFFE6B is deprecated; the permissible ma
+x is 0x7FFFFFFFFFFFFFFF. This will be fatal in Perl 5.28 at isnum.t l
+ine 12.
Use of code point 0xFFFFFFFFFFFFFFCE is deprecated; the permissible ma
+x is 0x7FFFFFFFFFFFFFFF. This will be fatal in Perl 5.28 at isnum.t l
+ine 12.
Use of code point 0xFFFFFFFFFFFFFFCE is deprecated; the permissible ma
+x is 0x7FFFFFFFFFFFFFFF. This will be fatal in Perl 5.28 in bitwise a
+nd (&) at isnum.t line 12.
Use of strings with code points over 0xFF as arguments to bitwise and 
+(&) operator is deprecated. This will be a fatal error in Perl 5.28 a
+t isnum.t line 12.
Use of code point 0xFFFFFFFFFFFFFE6B is deprecated; the permissible ma
+x is 0x7FFFFFFFFFFFFFFF. This will be fatal in Perl 5.28 in bitwise a
+nd (&) at isnum.t line 12.
Use of code point 0xFFFFFFFFFFFFFFCE is deprecated; the permissible ma
+x is 0x7FFFFFFFFFFFFFFF. This will be fatal in Perl 5.28 in bitwise a
+nd (&) at isnum.t line 12.
ok 3 - $z is a string, &~
ok 4 - $x is a number, ^
ok 5 - $y is a string, ^
Use of strings with code points over 0xFF as arguments to bitwise xor 
+(^) operator is deprecated. This will be a fatal error in Perl 5.28 a
+t isnum.t line 16.
ok 6 - $z is a string, ^
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and in perl 5.28 or newer it errors out like this:

$ perl isnum.t
1..6
ok 1 - $x is a number, &~
ok 2 - $y is a string, &~
Use of strings with code points over 0xFF as arguments to 1's compleme
+nt (~) operator is not allowed at isnum.t line 12.
# Looks like your test exited with 255 just after 2.
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So, don't rely on these bitwise methods where the arguments may be strings with high code points. Removing the 2 tests which operate on $z in this script allows it to run without errors or warnings in all environments.

FWIW I tend to agree with ig in Re^3: How to check if a scalar value is numeric or string? where he says "It might be better to revise your program so that it doesn't depend on such a subtle distinction."