What can I update my Imac to?

It depends on which version of iMac you have. To find out, click the Apple icon at top left, then choose About this Mac. In the resulting dialog, click the button for System Report... You will then see text next to Model Identifier like iMac 12,2. That's what you need to look it up.

My favorite operating system compatibility chart is the one on the MacSales.com website.

Can I update to something newer, like Mojave?

Yes. Your iMac should be iMac (21.5-inch, Mid 2014) and you can download Mojave from Mac App Store.

https://apps.apple.com/us/app/macos-mojave/id1398502828?l=zh&ls=1&mt=12

https://support.apple.com/en-us/HT201475

But, it is a major OS update. Please backup everything before you proceed.

https://support.apple.com/macos/mojave

This iMac isa 20" Early 2008...

Not sure what the hand off feature is.  New to Mac.

Handoff is where you start working on an email, message, note, contact, etc. on your iPhone and then complete the process on your Mac (or vice versa). It sounds really cool, but I don't use if very often personally--and your iMac won't support it anyway.

Next, you may be wondering how to get a copy of El Capitan and install it. Apple gives everything you need here: https://support.apple.com/en-us/HT206886https://support.apple.com/en-us/HT206886

I was able to install El Capitan and it worked perfectly.   Thanks