Your shuttlecock choice will depend on your playing level, what you feel when you are playing with different shuttlecocks and what you expect from them.
YOUR PLAYING LEVEL
Your shuttlecock choice will mostly depend on your playing level. There are different types of shuttlecock which suit different playing levels. Knowing the difference between two types of shuttlecock can sometimes be helpful!
When you start playing badminton, you simply need the shuttlecock to be strong so that you can make the most of your first few games. You will need a plastic shuttlecock with a foam head.
For regular practice you will need your shuttlecock to be even stronger but also to be comfortable on impact. As your game evolves so will your feel for the shuttlecock. Opt for a plastic shuttlecock with a cork head.
Once you have mastered badminton, you can move on to feather shuttlecocks with cork heads which will give you more control and comfort.
THE DIFFERENCE BETWEEN SHUTTLECOCKS
It may not be clear, but the difference between feather and plastic shuttlecocks, aside from appearance is that the trajectory of the shuttlecock will change. Take a look at this image which shows the different trajectories for the different types of shuttlecock. Line no. 1 shows the trajectory of a feather shuttlecock while line no.2 shows the trajectory of a plastic shuttlecock.