It is generally a good idea to wash the plastic before you apply paint. It is not always neccesary, but it can't hurt. I use dish soap and a toothbrush in warm water. Let the parts air dry after you wash them.
I use primer about 90% of the time, regardless of subject. Some people don't use it on aircraft or armor.
I am by no means a paint expert. I know there are a lot of them on this forum, maybe they can help you with the technical terms for what you are experiencing. I do know that temperature and humidity play a role in the way paint dries, though, so make sure you are not painting where there is a lot of humidity, and in an area that is about room temperature.
Also, the way you spray has something to do with it. You want to spray in several light coats, not one heavy one. Start with a mist that barely covers the primer. Do several coats like this, and finish with one or two good caots that kind of fill in the color. This will prevent the paint from pooling up on panel lines and stop it from running. It also preserves detail. This applies to the primer coat as well.
I know I haven't covered everything, but it's a start at lest. I'm sure someone can fill in where I missed.
Also, you can do a search on painting techniques. The airbrush question has been asked by many, many people on this forum, so I won't go into it, but rather tell you to do a search on it as well. Start with this section, and just run through the subjects, surely your question will be answered.
Good luck
BTW, welcom to the forums!