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 i 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.