iMac (2008) 10.5.4 - Prosc Speed 2.4GHz Intel Core 2 Duo - Memory 2 GB - 26 GB available storage space I have backed-up my HD with Time Machine to an external HD. I am ready to install upgrade to 10.6.3. After installation, do I need to run Time Mac

Yes. Time Machine will make a new backup once you have it setup in Snow Leopard. After updating to 10.6.3 you may wish to update to 10.6.8: Mac OS X 10.6.8 Update Combo v1.1.

I hope you don't really have a G5 iMac, because you can only upgrade an Intel machine to Snow Leopard.

And, you may like knowing:

Upgrading to Yosemite

You can upgrade to Yosemite from Lion or directly from Snow Leopard. Yosemite can be downloaded from the Mac App Store for FREE.

Upgrading to Yosemite

To upgrade to Yosemite you must have Snow Leopard 10.6.8 or Lion installed. Download Yosemite from the App Store. Sign in using your Apple ID. Yosemite is free. The file is quite large, over 5 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.

OS X Mavericks/Yosemite- System Requirements

Macs that can be upgraded to OS X Yosemite

1. iMac (Mid 2007 or newer) - Model Identifier 7,1 or later

2. MacBook (Late 2008 Aluminum, or Early 2009 or newer) - Model Identifier 5,1 or later

3. MacBook Pro (Mid/Late 2007 or newer) - Model Identifier 3,1 or later

4. MacBook Air (Late 2008 or newer) - Model Identifier 2,1 or later

5. Mac mini (Early 2009 or newer) - Model Identifier 3,1 or later

6. Mac Pro (Early 2008 or newer) - Model Identifier 3,1 or later

7. Xserve (Early 2009) - Model Identifier 3,1 or later

To find the model identifier open System Profiler in the Utilities folder. It's displayed in the panel on the right.

Are my applications compatible?

See App Compatibility Table - RoaringApps.

You are more than welcome.

BTW, in the future you do not mark your own question as Solved unless you actually solved it. Even though you gave yourself the Solved reward, you won't get the points, unfortunately. 😀