What do I do with Freighter and Supply Ships?

Your aassumptions are correct.In order to make survey ships you need to update your tech level.

freighter ships are sent to other civ's worlds to establish trade routs, which earn both civs credits per turn.

supply ships are sent to core worlds to give them some free manufacturing pts. i personally like to send 1 or 2 to a planet right before i turn it into a core world, so that when it becomes a core i can build a few entertainment districts in a few turns, which otherwise takes many turns because of low manufacturing from low approval.

freighter ships are sent to other civ's worlds to establish trade routs, which earn both civs credits per turn.

What are the requirements for sending to another Civ? Can I send it to any Civ I am friendly with in any way? Seems strange to just send a Ship to another Civs planet (out of the blue). 

pretty sure u can send one to any civ you're not at war with.

Freighter ships are used to establish trade routes with other Civs (as Basilisk83 said - as long as you're not at war with them).  Establishing trade with other Civs helps with their positive view of your Civ.  You can only send them to Core worlds.  I usually target capital worlds (esp. Minor Civ capital if close).  I've always assumed the higher the output values of the world I pick will yield a better result, not really sure if that's true   

Supply ships are useful for jumpstarting projects in the beginning of the game when your production is low.  They can be used to rush an important project when you don't have cash or control points.  Yes, you can only send them to core worlds, including the one that built the supply ship.  Later in the game you'll get a tech that allows you to build Industrial Goods which serve the same function, but can only be used on the planet where they are produced.

You can get more survey ships by surveying yellow anomalies.  That's a random thing so you could get other types of ships instead.  Right now yellow anomalies are not defended due to a bug the Devs are trying to squash, but eventually you'll need armed ships to defend you survey ship.  

The other thing that you need to know is that while colonies can't be managed they do serve a very important use.  They feed your core worlds with resources.  This can make core worlds very strong when they have several colonies sending them resources, so try not to get "Core World Happy Feet"

Thanks so much for the informative reply, scifi1950.

The fact that a civ can only make another Survey ship after finding yellow anomalies doesn't make much sense. I mean, the civ has already made one and thus ought to know how to build such a thing. If there is some sort of lore behind the yellow anomaly requirement, it certainly isn't explained. 

oh, anomalies arent required to build them, you need to research the right tech to unlock building survey ships, the yellow anomalies just have a small % chance of giving you a free survey ship after u survey them

Thanks for added info, Basilisk83 (and for the other replies above). Much appreciated.

Again, it doesn't make the gut check, right? If I have already made a Survey ship, why must I unlock the tech in order to make more?

np and yeah i agree with u on that one. and u have to research a few techs before u can research the tech that lets u build them, so it's def a little strange. some commander ships can survey too if it's listed on the ship. in fact all the Manti commander ships are survey ships with decent stats that level up (all survey ship and commander ships both level up btw)

Another way to use supply ships. Have a core world build a stardock, then just have that dock send supply ships to its own planet. As far as I can tell, your military and manu are completely separate (aka one does not consume from the other), so long term you are greatly increasing the manu available to a planet by feeding it supply from its own dock.