How to put site in 'Safe-Mode' to troubleshoot Drupal if site encounters problem?
First of all, I reccomend you install drush. It provides a ton of handy command line operations for Drupal that will allow you to do things if your site goes belly-up.


You should troubleshoot by turning off modules with drush
pm-disable. Look in your error logs to see if any modules in
particular are the culprit.
You can also enable and disable modules in the database, instructions
are here. You can also get some major issues if you move a
module to another location and don't rebuild the module registry,
which can be done with drush rr.
Make a backup of your DB (and in general, ensure backups are made on
a regular basis) before doing anything. Do NOT drop any tables or
delete any rows if you don't know what you're doing!


When doing updates, I would advise that you update each module one at a time in a development version of your site, using drush pm-update. Once the module is updated, run drush updatedb and also drush cc all, then check around the site quickly to make sure no major errors have occured.

Generally it's best to do core, then security updates, then version updates. It's also worth checking the version jump (especially with important modules such as Features, or Context) and making sure there are no drastic changes. As an example, Features 2 recently changed the way it stored field settings and it requires some clean up operations that will have to be done manually.

So take great care when updating modules!