Target disk mode, what cable do I need?

Theoretically a FW800 to TB2 adaptor and then a TB2 to TB3 adaptor should work. That’s considerably more than an external enclosure. Macsales.com has videos showing how to replace the hardrive of your iMac so that will show you how to remove the computer’s internal drive. Put it into the enclosure and you should be good to go. Once you’ve moved into your new computer keep your TM drive plugged in 24/7 and use the moved internal drive for daily or weekly clone duty. It isn’t a question of if you’ll ever need a backup but when, as you’ve learned. Sorry about the little lecture

See this Apple knowledgebase article: How to use target disk mode to move files to another computer - Apple Support

Great follow-up dwb. :)

Would the HDD USB box method work to setup the new Mac mini as the old iMac?

Yes it would allow you to migrate everything during the initial Mac Mini setup or after by selectively moving files from the old iMac drive to the new Mac Mini.

Thanks. I hate it when I see people going about migrating the hard way spending money on dongles they’ll only use once and may be dicey. Just like I hate seeing people migrating with WiFi. Until Apple stops changing ports/interfaces every few years a good backup or clone is the way to go.

when I give my backup sermon I always recommend a minimum of 1 TM backup and a daily or weekly clone - preferably 2 of each taking one set offsite each week. Personally for my desktop I have a two drive dock and 4 WD black drives, a pair at home and a pair offsite. One is TM backing up my media drive and internal drive, the other is a clone of my internal drive. I have amother backup in an enclosure of just data and multi-media. I’m anal about backups because in the late 80s I lost everything when my Quantum drive died of stiction and my floppy backup failed due to a single bad floppy. That left me with 6 classes of curriculum to redevelop from my partial paper records.

You could cobble together a bunch of expensive adapters and cables to get from FireWire to Thunderbolt 3 and hopefully be able to access the iMac's hard drive in target disk mode.

see > How to use target disk mode to move files to another computer - Apple Support

However it's much faster and easier to use the current Time Machine Backup from your 2010 iMac. (if you have one?...)

see > Restore your Mac from a backup - Apple Support

If you do not have a Time Machine Backup, then you might be able to migrate from one computer to the other using your local network. (OK for small amounts of data, but very slow for large amounts even using a direct Ethernet connection...)

see > How to move your content to a new Mac - Apple Support

Optionally you could remove the HDD from the 2010 iMac, put it into an empty USB box and then directly access or move your data from it to the new Mac Mini.

Agreed' backup drives or boxes and hard drives are cheap. Time Machine is OK for most users most of the time. It's better than nothing, but one should periodically check to make sure that it is up-to-date and working as expected. Personally I always preferred the daily, weekly, monthly CCC backup strategy and have for many many years. I did drop the daily CCC backups a few years ago and started running Time Machine backups on 2 different hard drives.