Time Machine Back Up to AirPort Time Capsule

First thing to confirm is to check to make sure that you do not have any kind of anti-virus application installed on the Mac Mini, or any other type of utility that claims to "speed up", "optimize", "clean", etc your Mac.

I've recently purchased a new Mac mini M1 and have been attempting to connect to my AirPort Time Capsule (3Tb) in order to start a new system of back ups.

Let's first make sure that that your Mac can "see" the Time Capsule using AirPort Utility. If it can't, backups will never occur. Open AirPort Utility on your Mac Mini, (located in the Utilities folder under Applications), click on the Time Capsule and then click Edit in the smaller window that appears. Next, click the Disks tab at the top of the next window to confirm that the Time Capsule disk......named "Data", unless you have renamed the disk.....appears. Post back on your results.

Assuming that "Data" does appear, the next check would be to try to run a backup of the Mac Mini as follows:

Hold down the Shift key on your Mac Mini while you click on the Time Machine "clock" icon at the top of the Mac's screen. Select Back Up With Consistency Scan to see if the backup will run. If it won't, post back with the exact text of the error message that appears on your Mac.

Time Machine is trying to delete enough data to allow the new backup to occur. This will seem like it takes forever.....and....the process may not be successful.

If you can afford to erase the Time Capsule disk and start all over again, things will go much more quickly that way.....and....you should have a lot of room for new backups.

I"m not a fan of trying to delete individual backups for each Mac, but you could try that as well. If it were me, and I wanted to keep the old backups of one or two Macs, I would copy the backups from the Time Capsule over to another USB hard drive......then.....erase the Time Capsule drive and start over again.

Chances are, you will never need the old backups, but some users find it very difficult to delete the old backups.

It's greyed out.

That....will need to be solved before progress will occur.. Try the following:

Power off the Time Capsule

Restart your Mac

Power up the Time Capsule and let it run for a minute or two

Try the process again

You may likely see "Preparing" for a long time.....even up to a few hours.....before the backup will actually begin. Let the process run until it completes or until you see an error message.

If an error message appears, post back with the exact text of that message please.