If you're looking for best practice for controlling change then you need to look at your service catalogue that ought to accompany such a thing.
You can then see what services are underpinned by the systems you're changing, and produce a standardised set of tests for each system, to ensure that they're both still working to your service level agreement.
Sorry to throw lots of ITIL jargon at you, but it sounds like someone's asked you about change management (and you yourself raise the ghost of 'best practice' for managing change) then this is what you need to do. Certainly a list of tests that I perform to determine that my network infrastructure and firewall are up to scratch won't help you much, because your priorities will not be the same as mine.
Whether you take any notice of the rest of this answer or not, the one thing I really do suggest to you is that "best practice" is to test functionality/availability from the perspective of the users trying to use a system; not to some narrowly defined test that might show a firewall as "up" because it responds to SNMP probes, while no one actually can use it due to a config error.
Once you've worked out what to test, it should be relatively simple, with most decent network monitoring software, to test, for example whether or not a URL on your website is available to the outside world, and flag an error if it fails or if it responds slower than you would like.