I'm looking for suggestions of common logs that must be checked on a Linux server. I know this depends on which services are running and so, but in general, which logs should I check?

Also, are there tools to make it easier? Or should I write my custom script?

Finally, how do you monitor the performance of a Linux server?

The 'default' log is /var/log/messages or /var/log/syslog. But you can log things to ANY file, so it all depends on how you have syslog set up. And each service can also have its own log file, independent of the system log.
Any tool you find will have to be customized, since only YOU would know what you're looking for, and where. Depending on your needs, it MIGHT be easier to write your own...it might not.
The same way you monitor it for any OTHER server. And again, it all depends on what's running on it. For some servers (Database servers, for example), it might be absolutely normal for a disk to be 99% full, and have most of the RAM consumed. Web servers may have MANY http processes running.

Your questions are fairly open-ended...not really a good answer for any of them.

Use monitoring software. Munin for example is simple to use.

Monitoring logs can be a rather complex task to undertake. It may appear simple at first glance, but when you start working on it, it can get cumbersome. i'll respond back shortly with a list of logs that are generally watched by every system administrator or application developer.

Log Files to Monitor on any linux host:

Ubuntu:
/var/log/syslog
/var/log/apache2/access.log
/var/log/apache2/error.log

RedHat
/var/log/syslog
/var/log/httpd/access_log (or access.log)
/var/log/httpd/error_log (or error.log)

I use <moderated> to monitor all my logs. However, to use it though, you have to have Nagios installed on your local server. if you dont want to install Nagios, you can use Cron.

<moderated>

The above log check command will basically check and monitor the [ /var/log/syslog ] log file for any new occurrences of the strings specified.

You may also want to keep an eye on your package install log files:

On Ubuntu, this log file is:

/var/log/dpkg.log (keep track of packages installed on the system)

On RedHat/CentOS hosts, log file is:

/var/log/yum.log

This tool is part of most distros http://linux.die.net/man/8/logwatch

(PS: Not every(!) system runs a webserver you know )

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