connecting apple mini to old i mac 24"

To upgrade from your old iMac to your new Mac mini, you'll want to first use an external drive and create a Time Machine backup on the old iMac. When that is complete, connect that TM backup drive to your new mini, startup the machine and follow the prompts. When asked if you'd like to transfer data, select the TM backup option.

Being that the iMac is so old, it's a good idea to only move user data and not apps or settings.

Back up your Mac with Time Machine - Apple Support

Transfer to a new Mac with Migration Assistant - Apple Support

No.

The Early 2009 24" iMac and the Mac mini running anything later than Catalina do not support Target Display Mode.

However you can use most any external display with the 2018 Mac mini.

Review the document: Use your iMac as a display with target display mode - Apple Support

Target display mode is a feature available with late 2009*-Mid 2014 iMacs. The display Mac has to be running macOS High Sierra or older and the signal producing Mac has to be running macOS Catalina (ca. 2019) or earlier.

* You say you have an "early 2009" model. :-(

It seems there is lack of clarity as to what you wish to do. If you wish to update the OS version on the iMac, upgrading a macOS depends upon the model Mac, specifications, and year. To get more information about your computer, choose:  (Apple menu) > 'About This Mac' in the upper left corner of any window, then "More Info..." or "System Report". There's more about this in "About System Information [Profiler] on your Mac" - System Information User Guide for Mac - Apple Support

Use the computer information, including the model year, and the links below to determine what is possible with your computer.

Click on this link to read how to get different versions of macOS --> How to download macOS - Apple Support

- The article also has links to system requirements for each OS version which you should check against your exact computer model. For any model produced in the last 5 years or so you should use the system requirements links for each OS version instead. This web site also has a reliable presentation of model and OS version compatibility: https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

- It helps to have more RAM (memory) than what is noted as "minimum" for any particular system or your computer may run slowly.

- You may find you must use Safari to get links for High Sierra and later using the Mac App Store application to open to a functioning page.

- Sierra, El Capitan, and Yosemite download as a .pkg. Click on this to have a converted version placed in your Applications folder as an "Install OS..." application. Run the installer and follow the directions.

- Upgrades may take some time (up to an hour or more) to install, with restarts and blank screens.

Before upgrading, back up at least your data in case something goes wrong. See this support document: How to back up your Mac - How to back up your Mac - Official Apple Support

Check if your old apps will still work with with the OS you want to install - https://roaringapps.com/apps This is particularly critical if you are upgrading to Catalina or newer, as older 32 bit apps will no longer work. To check for those, try this 32 bit application scanner application --> https://www.stclairsoft.com/Go64/

What external display do you have?

(brand and model #)