Do I need to upgrade my mac

From menu : About This Mac, what is the description in parenthesis, and specifically, the content in the red box for your Mac?

That will help you determine your hardware compatibility in the Check Compatibility section of How to upgrade to macOS High Sierra. You need a 2012 or later iMac to upgrade to Mojave.

It is not when chronologically that you purchased your iMac, but what is shown above by the red box. On upgrading to High Sierra, you will receive another year of operating system, Security, and Safari updates.

Netflix just requires the Silverlight plugin in your web browser to view on a Mac.

Firefox and Chrome are both compatible with Sierra and High Sierra.

No point on fretting if you can't get past Sierra or High Sierra.

In fact the current system you have on a 2010 Mac should be fine. Only Macs older than 2007 may have something to worry about:

https://help.netflix.com/en/node/23742

Your iMac can only upgrade as far as High Sierra, How to upgrade to macOS High Sierra - Apple Support

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.

If you want to confirm that you cannot upgrade to Mojave have a look at Section 1 here

How to upgrade to macOS Mojave - Apple Support