What's the easiest way to map the universe?

The method I personally use is to go to a busy sector like Seizewell with a shipyard and buy 10 cheap M5 scouts. Equip each scout with a scanner. Simply tell each ship to follow one of the many NPC freighters in the sector, and try to follow a variety of different races, as they will all have very different destinations. Your scouts will slowly fill out your universe map, though they will die when the NPC freighters pass through Xenon sectors. Also, they will not create a complete map, as NPC traders do not often go to the far-reaches of the universe and unknown sectors. The rest of the exploration can be done manually.

If you equip a ship with one of the special command or high level navigation software.

You can issue a command to it to explore, this will cause it to fly through gates till it runs out of sectors or most likely is destroyed.

Then you can set another ship to deploy your network.

In Albion Prelude there is Recon Bot and Explorer Command Software for ships. They both have command "Fly through gate".

Both of them have command "Map sector", but you need to order it for every sector.