Q: How do I update OS X 10.5.8 now that it is 2016!

More information about your iMac would be needed before someone could suggest anything.

If you're not sure, go to  > About this Mac > More Info > System Report. Copy all of the lines down to "Memory" (nothing below it), and paste it here.

Is your iMac compatible with El Capitan (10.11)? Check Upgrade to OS X El Capitan - Apple Support

What are the specs on your iMac? Please go to Apple menu > About this Mac and tell us year, processor and memory.

The first thing you'd need to do is upgrade to 10.6.8 to have access the the AppStore. But it depends if you meet the system requirements for upgrading further.

You have either the 20 inch early 2009 or 24 inch early 2009. Both are compatible with El Capitan.

You need, to first, uograde to OS X 10.6 Snow Leopard.

Before embarking on a major OS upgrade, it would be wise, advisable and very prudent if you have a good,working backup of your current system to an external connected and Mac formatted Flash drive OR externally connected USB, Thunderbolt or FireWire 800, Mac formatted hard drive. Then, use either OS X Time Machine app to backup your entire system to the external drive OR purchase, install and use a data cloning app, like CarbonCopyCloner or SuperDuper, to make an exact and bootable copy (clone) of your entire Mac's internal hard drive. This step is really needed in case something goes wrong with the install of the new OS or you simply do not like the new OS, you have a very easy way/procedure to return your Mac to its former working state.

Then, determine if your Mac meets ALL minimum system install requirements.

Mac OS X v10.6 Snow Leopard system requirements
Purchased Installer disc here.


http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard

To install Snow Leopard for the first time, you must have a Mac with:

An Intel processor
An internal or external DVD drive, or DVD or CD Sharing
At least 1 GB of RAM (additional RAM is recommended)
A built-in display or a display connected to an Apple-supplied video card supported by your computer
At least 5 GB of disk space available, or 7 GB of disk space if you install the developer tools.

Then use the OS X Software Update feature of OS X 10.6.3, located in the System Preferences Panel to update your iMac to OS X 10.6.8 and to updated all other installed Apple software and security updates.

Next,

You can chosse to install OS X 10.8 Mountain Lion (which is a paid upgrade) or the free OS X 10.10.11 El Capitán, you need one of these Macs:

OS X 10.8 Mountain Lion purchased emailed download code here.

http://store.apple.com/us/product/D6377Z/A/os-x-mountain-lion

Apple will send you an email for the special download code for the Mac App Store within three days of purchasing the download code.

iMac (Mid-2007 or later)
MacBook (13-inch Aluminum, Late 2008), (13-inch, Early 2009 or later)
MacBook Pro (13-inch, Mid-2009 or later),
MacBook Pro (15-inch or 17-inch, Mid/Late 2007 or later)
MacBook Air (Late 2008 or later)
Mac mini (Early 2009 or later)
Mac Pro (Early 2008 or later)
Xserve (Early 2009)
Your Mac also needs:
OS X Mountain Lion, Lion, or Snow Leopard v10.6.8 already installed
2 GB or more of memory (I strongly advise, at least, 4 GBs of RAM or more)
8 GB or more of available space

Next,
If you run any older Mac software from the earlier PowerPC Macs, then none of this software will work with the newer OS X versions (10.7 and onward). OS X Snow Leopard had a magical and invisible PowerPC emulation application, called Rosetta, that worked seamlessly in the background that still allowed older PowerPC coded software to still operate in a Intel CPU Mac.
The use of Rosetta ended with OS X Snow Leopard as the Rosetta application was licensed to Apple, from a software company called Transitive, which got bought out, I believe, by IBM and Appe could no longer secure their rights to continue to use Rosetta in later versions of OS X.

So, you would need to check to see if you have software on your Mac that maybe older than, say, 2006 or older.

Also, check for app compatibilty here.

http://roaringapps.com/

If you have any commercial antivirus installed and/or hard drive cleaning apps installed on your Mac, like MacKeeper, CleanMyMac, TuneUpMyMac, MacCleanse, etc. now would be a good time to completely uninstall these apps by doing a Google search to learn how to properly uninstall these types of apps.
These types of apps will only cause your Mac issues later after the install of the new OS X version and you will have to completely uninstall these types of apps later.
Once you have determined all of this, you should be able to find the latest versions of OS X by clicking on the Mac App Store icon in the OS X Dock and then login to the Mac App Store using your Apple ID and password and if you purchased a download code, input that code.
You can then begin the download and installation process of installing the newer versions of OS X from the Mac App Store.

Good Luck!

NSUDemon74 wrote:

 

Model Name:    iMac

[snip]

  L2 Cache:    6 MB

Please post the two lines after that from the same window as before, up to "Memory." This is important because several recent OS X versions require at least 2GB of memory, but probably at least 4GB would be required for smooth operation.

Yes. It looks like you're set on memory. The link in Duane's post above will explain the upgrade steps.

You need to purchase the 10.6 disc per my post.

Clicking on my link to the 10.6 Snow Leopard disc purchase will bring you right to where you need to begin.

You really needed to look at my posting as it is more conplete about many things NOT mentioned by Apple's own methodology.

Not that you gave me any credit.

Good Luck to you.

While you are waiting for your disc to be delivered, this would be a good time to get crackin' on everything you will need for backing up your entire computer system, especially, if you do not have a backup or recent backup already.

Good Luck to you.