How do I free up space for system files on an iMac with limited internal storage?

You appear to have current backups of all of your stuff, so i would suggest this:

Go to Time Machine settings and disable automatic backups. This will prevent TM from creating local snapshots that may be taking up space on your drive when the TM b/u drive is not available. About Time Machine local snapshots - Apple Support

If there are any snapshots taking up space on the drive, you can list them in the Terminal app using this command: tmutil listlocalsnapshots /

You may also see them listed in Disk Utility.

• In the DU menu bar, select View > Show All Devices.

• Select your "... - Data" volume in the sidebar, then from the menu bar select View > Show APFS Snapshots. It may take DU a minute (several?) to populate the list and make it actionable. When DU settles, you can select individual snapshots and click the delete button ( – ) below them to delete them. It may take some time for the OS to make the space available, even up to 24 hours. It's an SSD management thing.

Additional guidance: Deleting Time Machine local snapshots - Apple Community

Turning off automatic TM backups is not ideal, and neither is deleting snapshots. But your situation with this low capacity startup drive is already less than ideal.

Until you are ready to wipe the drive, reinstall the OS and not migrate your old files from b/u, this might get you moving forward.

The System Data can't be manually managed. What you can do is get an external drive formatted APFS for SSDs or OS X Extended (journaled) for HDDs and move your Photos and Music libraries to it as well as little used files in the Documents folder. It's wise to try to maintain a minimum of 80 - 100 GB of free space on the boot drive for optimum system and application performance.

If budget is a concern this setup is the most economical for getting an external SSD:

A 1TB is an excellent option (IMO). I have several. The 1.0TB Mercury Electra 3G 2.5-inch 7mm SATA 3.0Gb/s Solid-State Drive is currently on sale for $75.

If you don't free up additional space you could get to the point where you'd not be able to boot up and run at all.

Here's another possibility that you could consider.

• Purchase an external SSD such as a Crucial X9 Pro for Mac (1 TB, 2TB, or 4 TB) or one of the SSDs that Other World Computing sells. (The OWC site seems to be down at the moment, but I remember that they have USB, Thunderbolt, and USB4 40 Gbps drives.)
• Make sure that it is formatted as APFS.
• Install macOS on it. Then make it your startup disk, start up from it, and run Migration Assistant. Tell Migration Assistant to transfer your information "From a Mac, Time Machine backup or Startup disk" and point Migration Assistant at the internal SSD as the startup disk from which you wish to migrate. (Here, I am guessing that you can use Migration Assistant to copy your data to the new SSD. I haven't actually tried it.)
• Run off the external SSD. (But don't erase macOS from the internal SSD – you want to be able to boot from the internal drive if anything ever happens to the external one.)

Install macOS on an external storage device and use it as a startup disk - Apple Support

Transfer to a new Mac with Migration Assistant - Apple Support

Notes

• An alternative to a clean installation of macOS + a migration might be to use Carbon Copy Cloner or SuperDuper! to make a bootable clone backup of the internal SSD on the external SSD.
• A USB-C (USB 3.1 Gen 2) / NVMe SSD won't be as fast as your iMac's internal SSD, but likely will be fast enough, at a significantly lower cost than an external SSD in a Thunderbolt enclosure or a USB4 40 Gbps enclosure.
• If you get one of the Crucial drives, be sure to get a "for Mac" version. I read somewhere that the "PC" versions may not work as Mac startup drives "out of the box", and that applying the firmware update to make them work might require the use of a Windows PC.

Launch Disk Utility and check to see how many local snapshots are on your boot drive. You can erase all of them with the "-" button below them. I keep the latest one as it could be needed.

You bought a Mac with entry level storage capacity and are now discovering that 256 GB is extremely limiting. I never advise buy less than 512 GB, and would never buy less than 1 TB myself.

Yes, these space limitations are normal, especially with a low capacity startup drive.

Yes, it is possible to move your iPone backups and third-party (non-Apple) apps to the external drive.

Here are some things you can do:

• Empty the Trash in the Dock. 

• Empty the trash in the Photos app.

• Delete unused and unneeded application installers from your downloads folder and desktop.

• Reboot your Mac at least weekly.

• Transfer files that you don’t use daily to an external drive and then delete them from the startup drive and empty the trash. Files that take up the most room are movies, images and music.

This is some of Apple's guidance on the subject of freeing up space:

Free up storage space on Mac - Apple Support

You can try using OmniDiskSweeper to try & locate the large files & folders. Unfortunately there are areas even this utility is unable to see, but maybe it will give you a clue.

+1 on DI Johnson’s advice, you can also move large libraries (photos, music and movies) to an External Hard Disk.

Rather than trying to pare down the internal drive and use a combination of OS system on the internal and large data files on the external... can you just install the OS and everything else on one large external drive, and boot from that drive - with everything on it?

First, there is no reason to ever install or run any 3rd party "cleaning", "optimizing", "speed-up", anti-virus, VPN or security apps on your Mac.  This documents describe what you need to know and do in order to protect your Mac: Effective defenses against malware and other threats - Apple Community and Recognize and avoid phishing messages, phony support calls, and other scams - Apple Support.  

There are no known viruses, i.e. self propagating, for Macs.  There are, however, adware and malware which require the user to install although unwittingly most of the time thru sneaky links, etc.   

Anti Virus developers try to group all types as viruses into their ad campaigns of fear.  They do a poor job of the detecting and isolating the adware and malware.  Since there are no viruses these apps use up a lot of system resources searching for what is non-existent and adversely affect system and app performance.

There is one app, Malwarebytes, which was developed by a long time contributor to these forums and a highly respected member of the computer security community, that is designed solely to seek out adware and known malware and remove it.  The free version is more than adequate for most users.  

Unless you're using a true VPN tunnel, such as between you and your employer's, school's or bank's servers, they provide false security from a privacy standpoint.  Read these two articles: Public VPN's are anything but private and Security Risks: The Dangers of Using Free VPNs (eccu.edu).  

I recommend you uninstall Clamx A/V software according to the developer's instructions.

All the A/V software does is take up system resources, give false positives and lower bank accounts.

You should be able to recover individual files from Time Machine. You may have to hold down the Option key when accessing TM and select "Browse Other Backup Disks…" from the TM menu.