For example, if there's a new version of WP, as well as several plugin updates, does it matter which you do first? Just curious... and always trying to avoid messing up my website.
Not really, I'd always start at the top of the list and do the WP update first, then the plugins. It would be unusual if you had any problems.
Yeah it is unlikely to affect much. The ONLY way I can affect is that if the new version of a plugin requires functionality in the newest version of WordPress, then updating the plugin first would cause an issue. But that is incredibly rare. You can get it the other way round (WordPress removes functionality and an update breaks a plugin), but in all honesty most plugins would be aware of said change.