I was recently upgraded to 10.6.8 at the Apple Store but cannot Open or download Adobe Flash PLayer.Can anyone offer advice

Where exactly are you downloading it from? This is Adobe's latest flash player along with a drop down link to your particular OS version.

Just in case you are using the download from Cnet, it's always best to download direct from the developer.

Adobe - Install a different version of Adobe Flash Player

One other thing. You say you recently upgraded to 10.6.8 from the App Store. If you bought the DVD you have probably got the 10.6.3 version. (Check it in the  > 'About this Mac' drop down.) If that is the case, you will need to download and apply the Combo updater.

Mac OS X 10.6.8 Update Combo v1.1

And follow up with software updates to pick up the most recent security and iTunes updates.