Which is the best way to copy the contents of an old mac mini into the new one?

I like using a Time Machine backup.

Using Time Machine

Connect the two via FireWire, or if both have Thunderbolt, via a Thunderbolt cable and either use Setup Assistant during initial setup of the new Mac or Migration Assistant after initial setup. Note that if use Migrations Assistant a new users is created for the migrated files.

Move your content to a new Mac - Apple Support

How old is old? What is the old MacMini's model number? What OS X version is it running?