"$adminmail"

"$foo" is often very bad style. Consider it equal to '' . $foo . '', and just write it without quotes instead. Also, using single quotes allows you to write @ signs without escaping them (style advice: use double quotes only if you use their's specialties (read perlop)).
You use %mail just once, and could have put that in the sub itself.

In my style, this script would be:

#!/usr/bin/perl -w

# Well, to be completely honest: I don't like CGI.pm, and wouldn't use
+ it.
# I will for now, because I'm just rewriting :)
# You may have noticed I removed the tainting checks too.

use strict;
use CGI;
use Mail::Sendmail;

my $q = CGI->new(); # Updated as per particle's excellent observation

my $name    = $q->param('name');    # Note: used only once
my $email   = $q->param('email');   # Note: only used once
my $message = $q->param('message'); # # Note: couldn't choose ;)

# Config

my $adminmail = 'admin@yoursite.com';
my $from      = 'whoever@whatever.com'; 
my $subject   = 'enter subject here';

# End config

sendmail(
    To      => $adminmail,
    From    => $from,
    Subject => $subject,
    Message => $message
) or die $Mail::Sendmail::error;

# Note: redirects should be absolute
# (All modern browsers handle relative redirects, though)

print $q->redirect('thanks.html') or die "Can't find thanks.html: $!\n
+";
[download]

U28geW91IGNhbiBhbGwgcm90MTMgY
W5kIHBhY2soKS4gQnV0IGRvIHlvdS
ByZWNvZ25pc2UgQmFzZTY0IHdoZW4
geW91IHNlZSBpdD8gIC0tIEp1ZXJk