There are various log viewers for the desktop environments; gnome-system-log is searchable and user-friendly; as is ksystemlog for KDE (although of course you then need to install the KDE runtime). The gnome log should be the best one for you really, as it can be used to quickly filter the logs and find what you want; you can go straight to syslog, for example, and find the Ubuntu system messages and monitor how the system is working in the background with cron, etc. I recommend log viewing (and then researching things) as a great way of learning how the system works.
Terminal wise, you can watch any open logs in real-time with
tail -f <log file> and grep can be very useful to search for words such as 'Machine' 'Hardware' and 'error', which often pop up in faults: for example
grep -i 'machine' /var/log/kern.log. With grep it is important to use the i switch to ignore capitalisation, as you can miss important bits of the log without it. To look through a whole folder of logs, you can use
grep -ir <folder>.