Before Upgrading

When installing updates there are a variety of steps that should be taken first.  Properly preparing for the updates can relieve both immediate and long term stress and help keep your site running smoothly.

Staging

Before you update you should absolutely test the new software on a staging server.  A staging server is built exactly the same way as your live server, but no-one uses it for real business, so you can test new plugins, and new versions of plugins without worrying about crashing your live site.

Please read this extensive article on how to setup and maintain a staging server.

Backups

Before updating plugins a production server you should make a backup of your entire site.  There are a wide variety of options here. You could simply grab a backup of your database and FTP your files to a backup location.  A better option is to use a backup plugin or service. These make it much easier and faster to make a backup, which makes it much more likely that you'll actually do it.

Free Options

WordPress.org has many many backup plugins available.  They have a variety of options and tools, feel free to experiment and find one that fits you.

Paid Options

BackupBuddy is a plugin with an optional storage service that has a great many options and comes with paid support.

VaultPress is both a plugin and a storage service and is owned by Automattic.

Regardless of method, please be sure to always make a backup of your entire site before updating any plugins.