Noob Questions

You send that supply ship to a Core World.  A planet in which you've assigned a governor. 

As for the hexes. If you see a +1 Research or Manufacturing or anything, build that type of improvement on it. There will be other improvements later that you can build adjacent to those +1 hexes that will really boost the output.

Thank you that is so helpful.   I always thought you put the improvement next to a hex that had a '+ improvement'.     What does the supply ship offer to the Core world you put it next to?  

Supply Ships boost production of the building you are constructing at the Core World. You will see Stored goods show up on the Core world screen or you will see the amount of turns decrease on the building you are constructing. Like if I am building a galactic wonder that cost 2400, each supply ship generates 100 production so I could make like 24 supply ships from other core worlds and send them to the core world making the galactic wonder to speed up production.

1. What do you do with a Supply Ship?

Supply Ships provide a one-time amount of manufacturing, not an ongoing increase to manufacturing.  Generally, you would build them at a mature core world and send them to a new core world to help them get started.  However, I have had the shipyard at a new core world build supply ships that I send right to that new core world.  How you use them is up to you, but the main point is that they provide a fixed amount of manufacturing to the core world they are sent to.

2.  Do I care about the hex I put an improvement on or only the hexes next to where I put the improvement?  (paraphrased)

The simplistic answer is that the improvement will benefit from whatever bonus is associated with the hex where the improvement is built.  The more nuanced answer is that improvements not only benefit from whatever bonus is associated with the hex where the improvement is built, but also from bonuses offered by improvements built on neighboring hexes.  You really want to plan where you want to build improvements to cluster related improvements (manufacturing, research, etc.) together in the areas of the map that provide the best bonuses.  What you want to build will also be influenced by the characteristics of the civilization you choose to play.  For example, does your civilization need food?  Are they affected by pollution or crime?

P.S.  I am just an experienced player offering my advice.  I am not employed or compensated in any way by the creators of the game.

To clarify - you send the supply ship to the core world. Meaning it does NOT orbit or stay on the map. Set the destination of the supply ship TO end at the planet.  The supply ship disappears but you get 100 production.  Which really helps on brand new core worlds as those 100 production points go a long way towards improvements