1. The wiki designs its freighter as a civilian class ship. I figure that's probably because when I tried to make a military class freighter it had over 5000% chance of annual failure. So how will that work with transporting mines and cargo? Do I need to build something on the planet to load resources, or will they do that automatically? Do I need to build the mines at my production facilities on earth and then request a civilian transport to take them to Luna? Will I need to request a civilian transport to move the materials around?
Commercial vessels are important since they're the only kind civilian shipping companies can build (out of thin air: no resources nor wealth spent by the government). Said ships can only use commercial engines and sensors no larger than size 1. Obviously, no weapons are allowed either. CIWS appear to be the exception to that rule, though. Any violation to these rules reclassifies the ships as military. When you're designing this kind of vessels, you can check a particular box that hides all military-grade components, to make the process easier.
Freighters (along with other vessels like jumpgate constructors and asteroid miners) are best kept commercial since their cargo bays tend to make them pretty big, and building large military ships is a lot more expensive and time-consuming. Besides, civilian companies will likely be in charge of most non-military logistics, so you want them to have commercial designs at hand to use for that purpose. By default, civvie freighters generally ship trade goods back and forth between your colonies, and colony ships and luxury liners transport colonists around, frequently from your most populated worlds to the least populated ones.
When you want them to transport specific installations like mines and infrastructure, you've to set up civilian contracts using the Population screen. Set up supply contracts at the source of the number of items in question (which you'll have to build yourself), and demand ones at the destination. For instance:
[Earth] Supply: x1000 infrastructure
[Mars] Demand: x1000 infrastructure
The civilian companies should get to work then. As for the cause of their appearance, I'm not entirely sure, but it's tied to some extent to the existence of colonies besides your homeworld, and of course commercial ship designs (which have to be functional but needn't have left the drawing board). It can take a while. In my experience, it was a couple of years.
For shipping around minerals, I believe you need to use your own freighters, as civvie contracts don't seem to cover that. There's mass drivers, too, but I don't trust them.
2. How do I set up colonies. According to the game, the only nearby bodies I could colonize are Mars and Titan, and Titan will need a lot of terraforming before it's worth settling. Terraforming I get, you just need to build a terraforming building or have ships with a terraforming module, but I don't know how to set up colonies.
Go to the System View screen (the one with the sun icon), and that'll show a lot of information about the planets in a given system. Select one, and the option to declare it a colony should be among the set of buttons at the bottom of the window. The colony should then appear on your Population screen, along with the rest of your planets.
As for terraforming, make sure you send some infrastructure and colonists to the planet in question first, as the terraforming installations themselves need to be staffed, as opposed to other buildings like automated mines and deep space tracking stations.