How does trading work between your own cities?

Are you talking about Goods and Materials?

As settlements produce goods, wagon trains can carry them to sea ports, where ships can transport them to Europe. ref

So you could also move them in wagon train in another city, to turn them into finished goods. The Guns and horses could also be moved using a standard unit.

According to Steve Butts's review on IGN:

Our favorite addition, by far, is the automation of trade. The original Colonization let players create discreet trade routes and assign a specific wagon train to run that route, picking up and dropping off goods as directed. While it was a workable system, everything had to be done by hand, with the player setting up each individual pick up and drop off and having dedicated wagons ready for each strand in the network. Needless to say, it was a management headache that involved lots of scratch paper and frequent revisions when goods ran out and the system broke down. The new version lets players assign an import or export tag as well as stock limits to every single good in every single settlement. Then the player just has to set the wagon trains on automatic and they'll start redistributing goods on their own according to the tags and stock limits you've set.

 

You just tell the game that you want to export cotton where it's produced and move it to your weaving centers and the wagon trains take care of it. Of course, you have to have enough wagon trains to meet your trading needs, and you'll still want a couple under your direct control for one-time transfers, but the new system frees up part of your brain from worrying about the smallest details of your logistic system and lets you focus instead on your actual strategies.