Lately I have spent some time reading a book on Complex Analysis, namely the one by T. Gamelin. I have also looked at the Coursera course " Analysis of a Complex Kind " and so far I seem to be doing fine. What I am interested in, since I am self-studying, how should I learn ? I am not taking a course on a university, but I really like the subject and would like to delve deeper into it. I have received mixed advice on how I should study and was wondering, for instance, whether I should complete all the exercises and make sure I understand every proof or should I rather skim through it on my first read and go into details later on ? Also, do you have any recommendations on other resources I could employ for studying, such as specific problem sets or lecture notes you have found to be of interest ? Anything I should pay special attention to and make sure I understand as well as possible, for an example a specific proof or concept of crucial importance in the subject ? Any help is appreciated.
I feel the need to first address this part of your post:
or should I rather skim through it on my first read and go into details later on ?
The answer to this is absolutely NO. You will not learn the material nearly as well as you'd like the first time through, and by the time you're at the end (if you can even get there), it'll feel like a giant chore to go back through the material slowly.
On the other hand, trying to do every exercise in a textbook (well, most textbooks) is a huge challenge and time commitment, and shouldn't be necessary for the majority of subjects (maybe if we're talking Hartshorne or something the story is different..)
Rather, I recommend trying to find a balance between the two. Read proofs carefully, try to understand the big idea, and if you're completely stuck, don't be afraid to skip over it for the time being. Trying to learn proofs of some of the big theorems would be nice, but try not to blindly memorize proofs without understanding something. When you start working on a section's exercises, you'll find yourself looking back at definitions/propositions. Once you stop having to look back at the section, you probably know the material well enough to move on (but maybe do one or two more just to be safe). When you get stuck on a problem, ask for help here.
I don't really have an opinion on a good book. If you like the book you're using, just stick with that. If not, some standard references are Lang, Ahlfors, Stein. Or Brown & Churchill might be a good option for someone without much background.
You can start with an introductory text
You can use following reference book if you are keen in doing exercises
Visual complex analysis is a great book in understanding the geometry of complex function
and as mentioned by Alex, Lang, Ahlfors, Stein. Or Brown & Churchill are text text books of all time. However, as a beginner its better to understand why we need these variables? Their fundamental operations and geometry behind? For that purpose above listed book would be great start.