To make civilians ship something for you, you first need to have a civilian shipping line.
They will eventually form; If you want, you can subsidize them to expand those shipping lines.
Then, you go to the civilian/contracts page on a colony, likely your home, and Supply X.
This means whatever you supply, if it's actually available on the planet, is allowed to be taken and transported by civilians.
Afterwards, you go to the target colony, which in a standard start is quite often Mars, but can of course be any colony in any System connected with Jump Gates.
There, you go to the civilian/contracts page, and Demand X.
Now the civilians will start searching for those things, find them where you supplied them, and, if theres path from supply to Demand that doesn't lead through Unsecured Jump Points, they will start shipping that good.
You pay them with 'Wealth', an arbitrary good symbolizing the civilian economy.
You gain wealth by taxes, which can be your populations, Civilian mining operations, and civilian Freighters that freely trade consumer goods from one planet to another.
If you need a boost to your economy, you can research the matching tech (Expand Civilian Economy by 20%), or build financial Centers.
Overall, Civilians reduce micromanagement and produce wealth, but be sure to have your own transporter fleet for when it's really burning.