Q: First timer, 6 year old iMac, 10.8.6 w/4gb keep getting unsupported browser at gmail and our Netflix account home page has elongated graphics. Do we need some type of update or new machine?

Safari on OSx !0.6.8 Snow Leopard is now unsupported on many websites.  If your Mac Can support it you may want to update to a newer OS which will have a newer version of Safari.

Go into the Apple Menu->About This Mac, and tell us what model of Mac you have so we can give you options.

Based on the info, your iMac is a late 2009 model which is capable of updating all the way to El Capitan for free.

4GB is enough to run El Capitan,  however I suggest getting up to 8GB of RAM if at all possible.

http://eshop.macsales.com/shop/memory/iMac/DDR3_21.5_27

Before updating to el Capitan, make sure all the Apps you run are compatible with it.

https://roaringapps.com/apps

Then you can try to update to El Capitan:

http://www.apple.com/osx/how-to-upgrade/

Please do a backup before attempting to upgrade.