I manage a few servers for myself, friends and family as well as for the Libravatar project. Here is how I customize recent releases of Debian on those servers.
Hardware tests

apt-get install memtest86+ smartmontools e2fsprogs

Prior to spending any time configuring a new physical server, I like to ensure that the hardware is fine.

To check memory, I boot into memtest86+ from the grub menu and let it run overnight.

Then I check the hard drives using:

smartctl -t long /dev/sdX
badblocks -swo badblocks.out /dev/sdX

Configuration

apt-get install etckeepr git sudo vim

To keep track of the configuration changes I make in /etc/, I use etckeeper to keep that directory in a git repository and make the following changes to the default /etc/etckeeper/etckeeper.conf:

   turn off daily auto-commits
   turn off auto-commits before package installs

To get more control over the various packages I install, I change the default debconf level to medium:

dpkg-reconfigure debconf

Since I use vim for all of my configuration file editing, I make it the default editor:

update-alternatives --config editor

ssh

apt-get install openssh-server mosh fail2ban

Since most of my servers are set to UTC time, I like to use my local timezone when sshing into them. Looking at file timestamps is much less confusing that way.

I also ensure that the locale I use is available on the server by adding it the list of generated locales:

dpkg-reconfigure locales

Other than that, I harden the ssh configuration and end up with the following settings in /etc/ssh/sshd_config (jessie):

HostKey /etc/ssh/ssh_host_ed25519_key
HostKey /etc/ssh/ssh_host_rsa_key
HostKey /etc/ssh/ssh_host_ecdsa_key

KexAlgorithms curve25519-sha256@libssh.org,ecdh-sha2-nistp521,ecdh-sha2-nistp384,ecdh-sha2-nistp256,diffie-hellman-group-exchange-sha256
Ciphers chacha20-poly1305@openssh.com,aes256-ctr,aes192-ctr,aes128-ctr
MACs hmac-sha2-512-etm@openssh.com,hmac-sha2-256-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-512,hmac-sha2-256,umac-128@openssh.com

UsePrivilegeSeparation sandbox

AuthenticationMethods publickey
PasswordAuthentication no
PermitRootLogin no

AcceptEnv LANG LC_* TZ
LogLevel VERBOSE
AllowGroups sshuser

or the following for wheezy servers:

HostKey /etc/ssh/ssh_host_rsa_key
HostKey /etc/ssh/ssh_host_ecdsa_key
KexAlgorithms ecdh-sha2-nistp521,ecdh-sha2-nistp384,ecdh-sha2-nistp256,diffie-hellman-group-exchange-sha256
Ciphers aes256-ctr,aes192-ctr,aes128-ctr
MACs hmac-sha2-512,hmac-sha2-256

On those servers where I need duplicity/paramiko to work, I also add the following:

KexAlgorithms ...,diffie-hellman-group-exchange-sha1
MACs ...,hmac-sha1

Then I remove the "Accepted" filter in /etc/logcheck/ignore.d.server/ssh (first line) to get a notification whenever anybody successfully logs into my server.

I also create a new group and add the users that need ssh access to it:

addgroup sshuser
adduser francois sshuser

and add a timeout for root sessions by putting this in /root/.bash_profile:

TMOUT=600

Security checks

apt-get install logcheck logcheck-database fcheck tiger debsums
apt-get remove john john-data rpcbind tripwire

Logcheck is the main tool I use to keep an eye on log files, which is why I add a few additional log files to the default list in /etc/logcheck/logcheck.logfiles:

/var/log/apache2/error.log
/var/log/mail.err
/var/log/mail.warn
/var/log/mail.info
/var/log/fail2ban.log

while ensuring that the apache logfiles are readable by logcheck:

chmod a+rx /var/log/apache2
chmod a+r /var/log/apache2/*

and fixing the log rotation configuration by adding the following to /etc/logrotate.d/apache2:

create 644 root adm

I also modify the main logcheck configuration file (/etc/logcheck/logcheck.conf):

INTRO=0
FQDN=0

Other than that, I enable daily checks in /etc/default/debsums and customize a few tiger settings in /etc/tiger/tigerrc:

Tiger_Check_RUNPROC=Y
Tiger_Check_DELETED=Y
Tiger_Check_APACHE=Y
Tiger_FSScan_WDIR=Y
Tiger_SSH_Protocol='2'
Tiger_Passwd_Hashes='sha512'
Tiger_Running_Procs='rsyslogd cron atd /usr/sbin/apache2 postgres'
Tiger_Listening_ValidProcs='sshd|mosh-server|ntpd'

General hardening

apt-get install harden-clients harden-environment harden-servers apparmor apparmor-profiles apparmor-profiles-extra

While the harden packages are configuration-free, AppArmor must be manually enabled:

perl -pi -e 's,GRUB_CMDLINE_LINUX="(.*)"$,GRUB_CMDLINE_LINUX="$1 apparmor=1 security=apparmor",' /etc/default/grub
update-grub

Entropy and timekeeping

apt-get install haveged rng-tools ntp

To keep the system clock accurate and increase the amount of entropy available to the server, I install the above packages and add the tpm_rng module to /etc/modules.
Preventing mistakes

apt-get install molly-guard safe-rm sl

The above packages are all about catching mistakes (such as accidental deletions). However, in order to extend the molly-guard protection to mosh sessions, one needs to manually apply a patch.
Package updates

apt-get install apticron unattended-upgrades deborphan debfoster apt-listchanges update-notifier-common aptitude popularity-contest

These tools help me keep packages up to date and remove unnecessary or obsolete packages from servers. On Rackspace servers, a small configuration change is needed to automatically update the monitoring tools.

In addition to this, I use the update-notifier-common package along with the following cronjob in /etc/cron.daily/reboot-required:

#!/bin/sh
cat /var/run/reboot-required 2> /dev/null || true

to send me a notification whenever a kernel update requires a reboot to take effect.
Handy utilities

apt-get install renameutils atool iotop sysstat lsof mtr-tiny

Most of these tools are configure-free, except for sysstat, which requires enabling data collection in /etc/default/sysstat to be useful.
Apache configuration

apt-get install apache2-mpm-event

While configuring apache is often specific to each server and the services that will be running on it, there are a few common changes I make.

I enable these in /etc/apache2/conf.d/security:

<Directory />
   AllowOverride None
   Order Deny,Allow
   Deny from all
</Directory>
ServerTokens Prod
ServerSignature Off

and remove cgi-bin directives from /etc/apache2/sites-enabled/000-default.

I also create a new /etc/apache2/conf.d/servername which contains:

ServerName machine_hostname

Mail

apt-get install postfix

Configuring mail properly is tricky but the following has worked for me.

In /etc/hostname, put the bare hostname (no domain), but in /etc/mailname put the fully qualified hostname.

Change the following in /etc/postfix/main.cf:

inet_interfaces = loopback-only
myhostname = (fully qualified hostname)
smtp_tls_security_level = may
smtp_tls_protocols = !SSLv2, !SSLv3

Set the following aliases in /etc/aliases:

   set francois as the destination of root emails
   set an external email address for francois
   set root as the destination for www-data emails

before running newaliases to update the aliases database.

Create a new cronjob (/etc/cron.hourly/checkmail):

#!/bin/sh
ls /var/mail

to ensure that email doesn't accumulate unmonitored on this box.

Finally, set reverse DNS for the server's IPv4 and IPv6 addresses and then test the whole setup using mail root.
Network tuning

To reduce the server's contribution to bufferbloat I change the default kernel queueing discipline (jessie or later) by putting the following in /etc/sysctl.conf:

net.core.default_qdisc=fq_codel