Lecture notes, videos and other learning materials about $\infty$-category theory I've heard several times (and realized myself) that Lurie's tomes (extraordinary as they are) are not so ideal for self study.
I think it's a good idea to have some kind of compiled list of learning material about infinity categories (could be articles as well if they are considered to have a pedagogical style). Preferably they would have emphasis on intuition, and application rather than formality and rigor.
Lecture notes, Videos and any other relevant form of studying material would help a lot in travelling this technical terrain.  
 A: I found A Whirlwind Tour of the World of (∞,1)-categories by Omar Antolín Camarena (a student of Jacob Lurie) to be quite insightful for quasicategories.
A: Moritz Groth put up some excellent lecture notes: https://arxiv.org/abs/1007.2925v2
If you are more categorically minded, Emily Riehl's book has a lot about quasi-categories:
http://www.math.jhu.edu/~eriehl/cathtpy.pdf
and her website has a lot of exposition about cosmoi, an approach to $\infty$-categories via 2-category theory, and not requiring one to work in the setting of quasi-categories:
http://www.math.jhu.edu/~eriehl/
A: It depends on from what direction you are coming and in what direction you are wanting to go!  
As one of the developers of the basic theory from way back, I like to approach things via homotopy coherence as that is where the quasi-category approach comes from, and it also  more or less guarantees some nice diagrams and situations that are slightly more geometric or visual. Have a look at my n-Lab page: 
https://ncatlab.org/timporter/show/simplicial+foundations+for+homotopy+coherence
Other material that includes intuitive approaches include my book with Heiner Kamps: Abstract Homotopy and Simple Homotopy Theory, World Scientific, 462pp (ISBN 981-02-1602-5)
and the Cubo notes: Abstract Homotopy Theory, the interaction of category theory and homotopy theory, (survey article, updated version of lecture notes from summer school course at Bressanone), Cubo, 5 (2003)115-165, 2003)(and here: https://ncatlab.org/nlab/files/Abstract-Homotopy.pdf).
Those do NOT go very far but concentrate on the intuition and basic structure.  They may be too elementary for you but are intended to be readable. You can also access a version of the crossed menagerie: https://ncatlab.org/timporter/show/crossed+menagerie
which discusses a lot of the relationship with non-abelian cohomology, if that is what is of interest to you.  
I hope this helps.
A: There are now more resources since the question was raised. Markus Land's Introduction to infinity-categories grew out from lecture notes, and seemingly much better fitted for pedagogy.
On the other hands, due (or thanks?) to pandemic, there are more video resources. For example, there is a series of lectures by Achim Krause and Thomas Nikolaus recorded and uploaded on the YouTube channel.
