How can I update my iMac (21.5-inch, Late 2013)?

Okay your iMac from 2013 can upgrade as far as macOS Catalina 10.15.7.

You tried to install Sierra, this downloads as InstallOS.dmg and you opened that to get the InstallOS.pkg.

On opening that you would have got an installation window.

This does not install macOS Sierra but creates the Install macOS Sierra.app which you will find in your Applications folder,

have a look. Double-click on that to install macOS Sierra.

If you want to upgrade to Mojave or Catalina I would upgrade to macOS High Sierra first.

Follow the instructions carefully below, read everything.

For High Sierra, Mojave, Catalina,

Open Safari and click on this link, 

How to get older versions of macOS

Go to the Use Safari to download macOS section and click on the OS you want to install.

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.

Connect your mac to your router via cable rather than rely on WiFi.

Do not use your mac while it is downloading

and do not use any other devices that share the same internet connection.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS ******.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

If you are considering upgrading to Catalina be warned it does not support 32bit apps.

Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,

some updates may be free, some may be paid. Use this free app to discover which apps you have

are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see

if they are offering 64bit updates.

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal  operation of the OS and may even Corrupt the OS requiring a Reinstallation

It is sometimes normal for a mac to become sluggish just after installing a new OS.

This is because the mac is reindexing the internal drive and any other attached drives.

Depending on the amount of storage connected this could take quite a while.

To check if your mac is indexing click on the Spotlight icon in the menubar

enter a word, if it is indexing you should see a progress bar.

If the mac has completed indexing and is still performing poorly then try these three

procedures one by one.

How to use safe mode on your Mac – Apple Support (UK)

Safe Mode can take a long time to start up about 10 minutes so be patient.

When boo ing in to Safe Mode the mac is performing any disk repairs and clearing out caches.

Reset NVRAM or PRAM on your Mac – Apple Support (UK)

How to reset the SMC of your Mac - Apple Support

If your mac is still slow have you updated your third party apps to versions suitable for macOS Sierra.

Was your mac running well before you upgraded, if your mac was already having problems before often

upgrading the OS can escalate the problem.

Upgrading the OS should not affect the performance o your mac as long as the mac was healthy to begin with,

moving on to newer OS's like Mojave, Catalina and newer the mac benefits from having more RAM.

8GBS is recognised as being the minimum for newer OS's.

To check the health of your Disk download and run this app,

https://binaryfruit.com/drivedx there is free trial period.

Let us know how you get on.