You are going to see a WIDE range of opinions here, and that is cool. After spending a lot of time thinking about this I am going to suggest the book that got me into J-Lit some 10 years ago (this led down the path to where I am now, working on a PhD in Japanese literature).
Aside from Akutagawa being one of my favorite writers (and the namesake of Japan's most prestigous literary prize - The Akutagawa Prize), I think this particular collection is an excellent introduction to Japanese literature for a couple reasons:
The translation is really good. People know Rubin for his work with Murakami Haruki, but he has done a lot of other great stuff including translations of the woefully underappreciated Kunikida Doppo. It reads smoothly and I think he does a great job with some of the tricky, antiquated language that Akutagawa uses in the historical stories
It is a great blend of stories that focus on life in pre-Modern Japan (from Heian to Edo) & and Modern (Meiji/Taisho - 1890-1920s) so you get a good feel for the extended literary culture of Japan.
It includes some really famous stories. The most notable being "Rashomon" and "In the Bamboo Grove", the two stories that are the basis for Kurosawa's excellent film, Rashomon. Also, the story Rashomon itself is a play off a tale from the Heian-era text Konjaku Monogatari. Thus, you begin to really see the intertextuality of Japanese literature and film just from this one collection.
Lastly, it is an excellent introduction to some of the various literary forms in the Japanese tradition, including tales of morality ("Spider's Thread") and the autobiographical I-Novel ("The Life of an Idiot"/"Spinning Gears").
Ok, that is all I am going to say about this collection.
As a shout out to where my own personal research interests lie at the moment I would suggest everyone check out Islands of Protest: Japanese Literature from Okinawa - a collection of stories, poetry and drama from Okinawa, which holds a very unique and problematic position in the country. The experiences discussed in these pieces of literature really demonstrate what life is like in a neo-colonial state in the modern day. HIGHLY RECOMMEND IT!
Oh, I should also at least mention the inspiration for my username - Hayashi Fumiko's Floating Clouds (浮雲). My favorite text from one of Japan's most important 20th century female writers.