Question: I've had a few security problems with a server of mine because a few SSH users have been causing problems.

So I would like to:

* Track user logins and logouts
* Track activity of these SSH users, in order to discover any malicious activity
* Prevent users from deleting logs

Answer: 
The ssh daemon, `sshd`, has much of this built in and enabled. Here's are some sample lines from `/var/log/secure` on my machine (names and IP addresses changed):

```
Sep  7 08:34:25 myhost sshd[6127]: Failed password for illegal user root from 62.75.999.999 port 52663 ssh2
Sep  7 08:34:26 myhost sshd[7253]: User root not allowed because listed in DenyUsers
Sep  7 08:34:28 myhost sshd[7253]: Failed password for illegal user root from 62.75.999.999 port 53393 ssh2
Sep  7 11:55:18 myhost sshd[11672]: Accepted password for gooduser from 98.999.26.41 port 43104 ssh2
Sep  7 23:01:28 myhost sshd[22438]: Did not receive identification string from 999.56.32.999
Sep  8 06:31:30 myhost sshd[21814]: Accepted password for gooduser from 98.999.26.41 port 5978 ssh2

```

This example shows a couple attempts by somebody to ssh into this machine as root -- both were denied because root is forbidden. It also shows a successful login by the user named "gooduser". 

To fine tune *what* you see and in *which file*, read more in the [sshd\_config](http://www.openbsd.org/cgi-bin/man.cgi?query=sshd_config) man page -- specifically the options for LogLevel and SyslogFacility.
Following up on what Doug Harris answered (and only addressing part of your question - this seems to be how I work), the Logwatch package will email you a daily summary of a number of server logs, including the SSHd log. I get a summary of who successfully logged in via SSH, how many times, and from where, as well as what IPs tried to log in unsuccessfully and what credentials they used. It gets long if someone tries a brute-force SSH attack on a host where I'm allowing password authentication (I try to avoid that, favoring RSA keys instead, but the customer is always right and doesn't always understand public-key authentication. Anyway.)

To install Logwatch (which is basically just a collection of Perl filters for digesting various log formats) on Ubuntu, use `apt-get install logwatch` and then edit `/etc/cron.daily/00logwatch`, replacing `--output mail` with `--mailto you@yourdomain.com`. You'll get one a day. You can add more flags to tune which logs Logwatch actually reads.