Things I check before upgrading to a newer patch:
- Central Admin: Manage servers in this farm will tell you if a server needs to be upgraded. It may be that you are out of sync in in your patching levels, and Servers in farm page will tell you that. It also tells you the config database version, which is useful when selecting the correct patch.
- Central Admin: Check product and patch installation status, will give you the patching history success and failure. It also provides you with a very useful PowerShell script if you don't want to run the GUI version of Product Configuration Wizard.
- And if this is a new farm, I always run Product Configuration Wizard (PSConfig) on every server to make sure everything is OK before I start installing patches.
- Last, don't forget to download both the English version and any additional languages you have available in your farm.
Post patching, after running Product Configuration Wizard I check Central Admin: Check product and patch installation status, and that all web applications can be accessed from a client machine.