What upgrade path should I take on my iMac from Mojave to Monterey?

You're more than 3 behind my friend! :)

macOS Mojave 10.14.6 - Technical Specifications

General Requirements

OS X 10.8 or later

2GB of memory

12.5GB of available storage (OS X El Capitan 10.11.5 or later)*

Some features require an Apple ID; terms apply.

Some features require a compatible Internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Mojave:

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013; Mid 2010 and Mid 2012 models with recommended Metal-capable graphics cards)

https://itunes.apple.com/us/app/macos-mojave/id1398502828?mt=12

MacOS Catalina 10.15.7 - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

12.5GB of available storage (OS X El Capitan 10.11.5 or later)*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Catalina:

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013 or newer)

How to download macOS - Apple Support

https://apps.apple.com/gb/app/macos-catalina/id1466841314?ls=1&mt=12

macappstores://apps.apple.com/app/macos-catalina/id1466841314?ls=1&mt=12

How to upgrade to macOS Catalina – Apple Support

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

macOS Big Sur - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

35.5GB available storage on macOS Sierra or later*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Big Sur:

MacBook (2015 or later)

MacBook Air (2013 or later)

MacBook Pro (Late 2013 or later)

Mac mini (2014 or later)

iMac (2014 or later)

iMac Pro (2017 or later)

Mac Pro (2013 or later)

Learn about compatibility and how to identify your specific Mac model

https://apps.apple.com/us/app/macos-big-sur/id1526878132?ls=1&mt=12

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

Monterey…

macOS Monterey is compatible with these computers - Apple Support

• iMac – all models from late 2015, iMac 16,1 – 20,2
• iMac Pro – all models, iMac Pro 1,1
• MacBook & MacBook Retina – all models from early 2016, MacBook 9,1 – 10,1
• MacBook Pro – all models from early 2015, MacBook Pro 12,1 – 17,1
• MacBookAir – all models from early 2015, MacBook Air 7,1 – 10,1
• Mac Mini – all models from 2014, Mac Mini 7,1 – 9,1
• Mac Pro – all models from 2013, Mac Pro 6,1 – 7,1

https://wpbeaches.com/apple-minimum-system-requirements-for-macos-monterey-12/

If I were to skip any it'd be Catalina/10.15.7

Just make sure you have a good backp or clobe or 2 before starting.

To add from previous Contributor.

Notation regarding Monterey Specifically.

If there is an After Market Drive in this computer, there will be issues installing Monterey.

There is an EFI Boot ROM included in the Monterey Installer.

It will verify if there is an Apple Original Drive or Not.

If it determines there is an After Market Drive - it will refuse to install period.

Background.

The EFI Boot ROM is required to Flash the Computer and Not the Drive.

Once the Flashing of computer is achieved, the installation will continue.

The solution, if applicable to this machine, is to swap Out the Drive and replace with Apple Original.

Run the Monterey Installer to that drive

It will satisfy the Flashing of computer

Reverse the previous swapping of drives with the After Market Drive

Run the Monterey Installer to the drive and it should function

Mojave was the last macOS that ran 32-bit applications, and before you do anything else, you need to understand what 32-bit software you may be currently dependent on. Download and run the free Go64 application as it will detect all 32-bit applications that you have currently installed. This application is self-contained and without an installer. Applications like Office 2011 for Mac, any of the Adobe CS? suites, Apple's iWork '09 suite, iMovie '09, etc, will not function on Monterey. Your documents would not be affected by Monterey.

So bottom line, you may be facing some new third-party application expenses upgrading to Monterey, and certainly the need for 64-bit versions of those apps that are stated by their vendor to be compatible with Monterey. Same issue with your printer/scanner vendor driver support. If no Monterey or Apple AirPrint support, then paperweight.

If your iMac was purchased with an internal Apple Fusion drive, it will be wickedly slow-running Monterey as current operating systems are designed for SSD, not hybrid SSD/HD solutions like Fusion. If that is the case, I suggest an external fast SSD in an enclosure to install Monterey on and reliable sources are Otherworld Computing or Crucial.