An alternative method to leveling up a lot of your crew skills is by sending your companions out on missions to gather different materials or items using your crew skill. Missions require nothing on your part besides a few simple mouse clicks (and some credits) and are extremely rewarding. Technically you could maximize your crew skills by just doing missions without ever leaving town a single time. To view the missions available to you click the diamond icon at the very top of your screen to open the crew skills window.
The window shows you what crew skills you have available, what their levels are and also shows you a list of all your companions. From here in order to select or view any missions currently available to you, you must click on the picture of the crew skill next to your companions' that you would like to run the mission for. Doing this will bring up an entirely new window with even more options. A picture of this window and an explanation of its contents can be found below.
Usually you have a great deal of different missions to choose from, depending on what your skill level is and what level range you are looking through missions for. The picture above shows a lower level range and a lot of different missions available in that range. Before I explain how the missions work and how you know you're choosing the right one, I'd like to explain a few other things. First, the color of the mission tells you what the mission level is, second the time next to the mission tells you how long it will take your companion to complete the mission. Finally to the right of the time is the cost of the mission, telling you how much it will cost for your companion to complete the mission.
Below that it tells you about the mission. This is just background information and lore, it serves no purpose besides being just that and does not effect the mission at all. Further below that is the next part I want to talk about, but first I want to mention what is at the very bottom of the window. Under where it tells you what companion you have selected (Qyzen Fess in this case) it tells you what the extra perks this companion has towards crew skills. As you can see Qyzen Fess has +15 Archaeology and +5 Bioanalysis.
Now for the important part that will allow you to collect all the materials you need for your crafting skills if you use the information correctly. Below is a better picture of what I am talking about so I can easily single it out.
At the very bottom of this picture see where it says "Moderate Yield: Grade 5 Biochemical Compounds"? This tells you what items you should expect to get through this mission. Obvious, right? I know. The thing that matters most about this though is that no matter what level you are you will always be able to gather all the materials back to Grade 1. In order to change what level range you want to gather from just change the level bracket in the upper right hand corner of the window, choose what Grade material you want and... profit!
Something else I would like to add is not all of the rewards will give you a "Grade xx" item, some of them give you additional things such as First Aid Kits, Hypo-syringes and such for Bioanalysis, Augments for Slicing and even more for other crew skills. That's about all there is to doing missions with crew skills!