How can I discover my weaknesses in chess and order them from biggest to smallest? I'm guessing a strong GM could analyze a couple of my games and do this for me. But I'm not sure it's worth the economic investment. Is there another way for me to do this?
Made two self-assessment book recommendations to a related question (How to automatically evaluate a player's performance in a game?). Igor Khmelnitsky's Chess Rating Exam is pretty thorough at evaluating every aspect of your game.
BTW, really like that you are approaching this in a analyzing/organizing your "biggest weaknesses" manner rather than embarking on the fairly commonplace it must be the openings, so I need to study openings knee-jerk/inefficient study plan.
This is a matter of your personality. If you are a type of person who can objectively assess own weak and strong sides, then the same skills can be used for chess. In this case you need to analyze your games, which part of games are your weak sides, in which openings or end games you feel uncomfortable or good, whether you prefer tactical type of games or positional and how you manage to deal with them. Just do two columns for pros and cons then you might understand you weknesses and strengthes.
However if you can't really assess yourself then help of a GM or IM can be handy! And, it will be handy anyway :)
To discover your strong and weak skills requires you to also understand what types of skills that are relevant for your chess strength. I think the process of understanding the complete skill set is a gradual process over a long period of time. Having said this, you don't really need to know all your strong and weak skills in order to improve. Instead, you can focus on one skill at the time. For instance, calculation of variations, rook endgames, positional play, etc. Pick an area that you think will have a clear positive effect on your results and start working with only that skill. Of course, the ideal situation is when you have a skilled mentor that has a deep insight in your skills and guides your development. It is indeed rare to find such a mentor free of charge. If you think that improving in chess is important, consider finding a mentor.
As @Andrew pointed out you should look at your previous losses (and blown wins), and find the root cause. Try to be as honest as possible with yourself (this is easier said than done!). You might want to do this for all your losses, maybe make a spread sheet of all your losses, with an illustration of crucial position. You might start to see patterns that you were not aware off, for example, you might see that you lose all your games in even knight endgames, or it turns out that you lose all the games against the Scandinavian.