Pretty self explanatory; when you stop at a space station go to 'starport services', then the bulletin board and select a mission. I find the easiest ones to be the transport cargo missions. The icon will be a piece of cargo with a white arrow beside it. The arrow means that they will give you the cargo and you just take it from where you're at to the destination listed. You don't have to worry about going out and finding the cargo that way like some of the missions without the arrow beside it.
Just do 5 different missions of any type and it will unlock.