Ask the community
As the hundreds of responses to my article show, there's a vibrant, active Linux community willing to offer advice to those who need help. One good place to go for advice about Ubuntu is ubuntuforums.org
Don't think of Linux as Windows
Numerous people wrote to say that for installing and updating software, it's a mistake to think of Linux in the same way as one thinks of Windows. In Windows, you often install software via a self-contained installer file. Seth Rattan was particularly helpful in explaining that Linux uses a repository system in which software necessary for installation or update is stored in an online repository, which your version of Linux then contacts. At that point, installation or update should be simple and automated. More on this when I write a follow-up article.
Know your distro
Unlike in Windows, you'll generally need to install a version of software specific for the Linux distribution you're using. Once you know that, you can seek out software specifically written for your distro. As "Luis" puts it succinctly in a comment: "The Ubuntu default package is .deb. Download anything with this extension and Ubuntu will install it. Just click on it."
Use the Synaptic Package Manager
When looking for software updates, I looked to the Update Manager, assuming that was the best place to find updates. But several people explained that in fact, a better bet is to use what's called the Synaptic Package Manager, available via System-->Adminstration-->Synaptic Package Manager.
Use Add/Remove Applications
Several other people said the best way to install new applications is to use the Add/Remove feature, available via Add/Remove, rather than browsing the Web and trying to download files that way. I tried it. They were right.
Be careful when using Update Manager
Update Manager doesn't distinguish between important and unimportant updates. So when you use it, the number of updates you see can be overwhelming, particularly for someone new to Linux. So don't get overwhelmed, and think you won't be able to update anything when you're confronted with such confusion.
Re-think the application numbering convention
In my article, I complained that I wasn't alerted that a 3.0 version of OpenOffice.org was available, even though I was using the older version 2.4. "Andy" explained why that happened:
In the case of OpenOffice, it's not "OpenOffice" 2.4 and 3.0, it's "OpenOffice 2, revision 4" and "OpenOffice 3, revision 0," no different than Windows XP SP2 and Windows Vista. Expecting [Linux] to update from one to the other is akin to expecting Windows Update for XP to automatically recognize that Vista is available and 'update' the system to use that.
These tips are just the start --- I'll have more detailed updating and installation information for Linux newbies in an upcoming article.