If you buy a new car and never change the oil, it will eventually be ruined. Computers need maintenance too. But you have to be the mechanic.
Abraham Lincoln said that if he had eight hours to cut down a tree, he would spend the first six sharpening his ax.
When applications (e.g. Firefox, Word, Photoshop) pop up an "updater" box for a minor (free) upgrade, accept it: most of these are security fixes or fix crashing problems.
Adobe Flash has a feature that will automatically update if a new version is available, starting with version 11.2 (March 2012). Flash is used in a lot of web sites nowadays, and has been the source of many malware infections. Turn the auto update feature on.
When Apple Software Update wants to update your software, these updates often fix crashing bugs or security problems. Apple provides updates to the operating system, and to the applications that run on your computer. Often, these updates are only provided for the most current version of the OS, so that's a good reason to stay up to date. Select to check for updates. This will launch the Apple App Store and show the Updates tab.
Usually it is safe to update Apple applications without any special precautions.
When a printer or scanner manufacturer releases an update for a device driver, this software is usually made available by Apple via software update. Often, there is a delay of weeks or months: Apple seems to batch such updates. If you have a device already installed, Software Update should notice it and update your driver. If you buy a new device, and plut it in and then add it in, Mountain Lion should check whether it has the latest driver and invoke Software Update if necessary.
Sometimes, for updates of Mac OS X, it is a good idea to wait a few days and watch the comments in MacInTouch to see if others are having trouble; this kind of trouble may affect older computers more than recent ones. I usually wait a week or so before installing OS updates, and watch MacInTouch for complaints. When I do install, I make sure each computer is fully backed up, and then run Disk Utility to repair permissions on the disk. I then eject any external disk drives and disconnect devices other than the keyboard and mouse. Some users have found that it is safer to download the "combo updater" and apply that, rather than just saying "install" to Software Update.
When Apple Security Updates are published, don't put them off too long.
Some applications (e.g. Adobe Photoshop) release major new versions every year or so. Usually these upgrades are some fraction of the cost of the new application. Sometimes you don't need to buy them: they add features you can do without. I have been buying about every other new version of Photoshop and Illustrator. When you are getting ready to move to a new version of OS X, you may have to upgrade applications first.
Every few years, Apple will come out with a new major version of OS X, usually a paid upgrade for about $100. Don't rush into upgrading: often it is wise to wait for the .1 or .2 version. But do upgrade eventually; Apple stops supporting old versions with security patches after a while. Also, if your OS is too old, you may find yourself in the situation where you can't install some new application, because it requires interfaces from a newer version of MacOS. When you upgrade the major OS version, sometimes this means that you have to upgrade some applications as well.
I held off on switching from OS X 10.4 "Tiger" to OS X 10.5 "Leopard" for almost a year, because I was using a programming tool (Eclipse) that wasn't ready for 10.5. Finally I was able to make the change and I'm glad I did: Leopard was faster, had nicer features, and supported Time Machine. Mac OS 10.6, "Snow Leopard," was available for $30 in fall of 2009; I waited almost a year to install it. I skipped 10.7 "Lion" entirely, because none of its features helped me; I installed 10.8 "Mountain Lion" six months after it came out, and it has worked OK for me.