Your Mac is a "iMac (24-inch, Late 2006)". Your Mac can only support up to Mac OS X 10.7.5 Lion. 10.8 or latest version require imac 2007 or later model.
The problem is, latest or a compatible version of Safari or other browser requires Mac OS X 10.9 or later. That means, even though you can upgrade to 10.7.5 Lion, it is not worth it and doesn't fix the Safari issue. You still will be using an outdated OS X.
The latest version is macOS 10.12 Sierra. I'd recommend to get a new iMac because it is not worth upgrading this machine due to incompatibility and obsolete hardware.
Re backing up, you can buy any model of external hard drive and backup the entire Mac using Time Machine backup application. Time Machine is an application that can backup the entire Mac easily.
Just buy an external hard drive from any local store, then connect it to your Mac and go to Apple > System Preferences > Time Machine. Then select that hard drive there and it will do a full backup (it will take a few hours to do a full backup). Then you can restore this entire backup to your new iMac when you get it easily (while setting up the new Mac, it will give you an option to transfer data from Time Machine backup or old Mac). Se this article --> https://www.uta.edu/oit/cs/software/apple/how-to/Enable%20Time%20Machine/index.php
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