I’m not sure how much time and effort you are able and willing to spend on your project. If you really want to get to grips with the philosophical side of existentialism, you should try to get the gist of Heidegger and Sartre. In philosophy I think Sartre is not as well regarded, but it would be a pity to ignore him. Besides those two there are a lot of others, but that might be for later study.
For understanding Heidegger I’d suggest first reading a bit yourself and failing miserably to make any sense of it, but it might give you a taste of what he’s doing.
Fairly accessible are
- The origin of the work of art
- The question concerning technology
If you are sufficiently well read in philosophy, you might try the beginning (first 40 pages or so) of Being and Time, where Heidegger does try to take the reader along his thought process.
- Introduction to metaphysics
- What is metaphysics
- Identity and difference
You might try to leaf through them to get a but of the flavor. Personally I found a lot of his writing to be better appreciated like poetry, where he forces you to consider the implicit connotations of the words we use, and whether those have deeper meaning. I’m not sure how well that holds up in translation.
For an introduction to his actual thought I found R.C. Solomon, From Rationalism to Existentialism, very helpful. He does require you to follow the actual philosophical development, but presents this in as accessible a way as possible. Arguably he also simplifies a bit and may be incorrect in places, but at least it gives you a good basis for understanding the stakes in the debate. From a different perspective you could look at D.E. Cooper, Existentialism, which also is a fairly accessible text at the level of philosophy majors.
If you are willing to go in depth into Being and Time, you can read H.L. Dreyfus, Being-in-the-World, along a close reading of Being and Time. That is a close-reading commentary of the first part of the book.Highly rewarding but it takes time.
As regards Sartre, unfortunately there is not too much to him. After reading introductions such as the ones I mentioned above, you should at least read the introduction of Being and Nothingness, which should then be fairly easy to follow. From the main body of the text you could focus on selected parts that you are particularly interested in, either the more ‘metaphysical’ chapters or the more ‘moral’ chapters (but my current memory is a bit hazy). They were a bit long-winded so you might not dwell on every word, instead trying to follow the broad outline of the argument. If you are really enthusiastic of existentialist philosophy there are actually a number of minor works by Sartre which are interesting, but then we are far outside the scope of an introductory course. ‘Existentialism is a humanism’ is an easy read but doesn’t give you much understanding of what existentialism actually is, philosophically speaking.
Sorry for the slight rambling nature of this post.