It's early in the game for me yet, and I have just run out of resources way too fast on my homeworld. My other planets don't have ships yet. Can someone tell me how to send ships from a planet to pick up resources from another planet and carry them back (the method for sending ships to drop resources off is apparent to me) ?

You can move your ships around.
i.e. Move your ships from planet A and station them to station B.
It's the deploy mission.

But you there is no pick-up mission.
You have to manually deploy them.

Deploy 'em, then once their there, have 'em transport back, right? Or even re-deploy back home with more resources on board - which is a technique you may have to use a lot, since you often want a ship to transport resources and not come back; especially if you want to swap resources, having some coming to the originating planet on the return trip. I might have figured this out, but for the fact that running a Deploy mission results in an outgoing mission report that includes an arrival time, and a return time.

I think you figured it alright.
But there is no return time for the deploy mission. Unless you recalled them.