Which books should I read and in which order to learn and understand (existential) philosophy more?

If you want to start building a background then Bertrand Russell's History of Western Philosophy would be my suggestion. In order to understand existentialism you also need to understand the traditions from which it arose.

Quoting Pantagruel

If you want to start building a background then Bertrand Russell's History of Western Philosophy would be my suggestion.

This completely sidesteps existentialism. Russell didn't consider it important.

As an intro, the primary sources can be pretty hard going, but Norton have a book of selected key primarytexts that's useful. It's a bit unkind to Nietzsche, focussing more on his philological justifications of racism than his will to power, but then I don't think Nietzsche is the best place to start with existentialism anyway.

To get a feel for it, two literary texts:
Camus: The Stranger
Sartre: No exit

And:
Camus: The Myth of Sisyphus

Quoting Kenosha Kid

This completely sidesteps existentialism. Russell didn't consider it important.

It certainly does, since the book was written while existentialism was essentially in the process of self-definition by Sartre. As I said, a good way to establish a background. It still covers Schopenhauer, Nietzsche, etc. I think building up a context for existentialism that predates the explicit self-formulations of existentialism is a sensible and balanced approach.

I would also not recommend Russel. He is very much into the analytic tradition, which is not where existentialism comes from. I myself was taught from a rather down to earth book. The Penguin History of Western Philosophy by D.W. Hamlyn. A great intor is also to take a philosophy course. It is handy to have someone around to throw ideas at and recommend you the specific works you might be interested in. Schopenhauer is good too as it is rather easy to read and most of all fun, because he wrote well and he is interesting in his own right especially if ou enjoyed the mystics like Plato Meister Eckhart etc.

I've read almost everything Nietzsche published, listened to a course on him, and read Kaufman's superb guide. I would say if you're going to read one to start, go with Beyond Good and Evil. Generally, Nietzsche's mature thought is considered to begin with Thus Spoke. Thus Spoke is a difficult, more artistic work, and it's easy to misread with no background in Nietzsche. Beyond has his bombastic, highly enjoyable aphorisms, but also lays out his philosophy in much clearer terms. Geneology flushes out the thoughts in Beyond in a more concrete format, but I found it drier. Kaufmann is a great resource for Nietzsche, going over his thought and personal life.

It's worth going back to his early works to see how he developed his final thought. If you've been reading the Greeks, you will find the Birth of Tragedy interesting. It gets skipped often because it is his first work, and far from his mature vision, but you get to see how he is diagnosing problems. It's also significantly different from later works, which makes it fresh. There is a great take on Hamlet in there too, which is super short, but poignant.

It's worth also reading Notes from the Underground. Dostoevsky is a titan of Christian existentialism in his own right, but he read Nietzsche and Notes has the benefit of being a response and much shorter than his other work. The Brothers Karamazov is my favorite novel and is also a great book on existentialism.

Camus' works benefit from being short. I would say The Myth of Sisyphus is the best to start with as an example of existentialism. The Stranger is good too. The Plague is looking more into politics.

If the religious side interests you, Dostoevsky and Kierkegaard are good placed to turn.

Being and Time is another hugely influential work. Probably the most timely as philosophy has found a new practical niche for itself in neuroscience/cognitive science. I found it to be interesting the frame of the Hard Problem of Conciousness.

Quoting Tobias

I would also not recommend Russel. He is very much into the analytic tradition, which is not where existentialism comes from.

If you really want to understand something, you need to understand what it is not, and where it came from. The first two philosophy books I ever read at age 17 were Being and Nothingness, and Beyond Good and Evil. I wish that I had taken a more balanced approach then. All authors have a perspective. On the whole, I find Russell to be reasonably balanced, and an excellent writer. If you only ever read what you prefer, you will quickly develop myopia.

Is it a good idea to start Nietzsche by reading Beyond Good and Evil or what should be the first book I read by him?

Less solipsistic – subjectivist – descriptions of 'existence':

Critical Existentialism, Nicola Abbagnano
The Ethics of Ambiguity, Simone de Beauvoir
Reason and Existenz, Karl Jaspers
Tragedy and Philosophy, Walter Kaufmann
Joyful Cruelty, Clément Rosset

I started Nietzsche with Beyond Good and Evil, my phil. prof recomended me to do so and who am I to object... so I would still say yes to your question.