I'd say just pick whichever one gets your attention the most and just go for it. The songs, anime, manga, and novels are all different timelines anyway, so it doesn't matter too much which one you choose first. If you like the series, you'll eventually read/watch everything in the end anyway.
For me, the order was roughly songs -> manga -> anime -> novels (still haven't read the novels yet). But there was some overlap; I'd only listened to a few of the songs when I started the manga, and I got through the rest of the songs afterwards.
-
The songs are good place to start if you have a short attention span or you don't want to get into anything too hefty to start with. You can watch a few songs whenever you have a bit of free time. There are lots of utaite covers and Nico Nico Chorus covers of the songs, so there's variety. You can usually find subtitled reuploads of these music videos on YouTube, too. (Watch out for spoilers in comments.)
This was my first song too. Don't worry about going in order; they're Vocaloid songs, so each music video is relatively self-contained. Just choose the most interesting-looking ones to start with, like you would on a typical unproductive YouTube walk. I dunno about the rest of you, but I love a good achronological experience. Once you've watched more of the music videos, the bigger picture should coming together.
... - (Playlist of songs in order by release date: https://www.youtube.com/playlist?list=PLc3ovJ5PZjKRleiWwSlIBMtql8nYISZaJ )
... - (Playlist of songs in chronological order by plot: https://www.youtube.com/playlist?list=PLmjr2gQDAIUsTf1Fbz7wZulDEilvAr3JF )
-
The manga is the most straightforward experience. If you were confused by the songs, the manga makes things more clear. I personally really like the way the manga portrayed the Yuukei Yesterday arc.
-
The anime is animated by SHAFT, so it might be hit or miss for some people. The general atmosphere feels somewhat stylized and experimental. Some people don't like the anime, but I thought it was okay. The animation quality in episode 9 was horrifying, but they fix it in the BD version, so if you can, try to get that version. Also, the anime has many of the songs and also some screenshots from the manga in it.
-
The novels: ...well, I haven't read these yet, but it too adds something that the other forms of media doesn't, e.g. more detailed specifics about Kido's backstory. If you like reading words, this is probably a good choice. Jin (the songwriter for all the KagePro songs) is the author for these novels! (Jin also wrote the script for the anime, and is in charge of the story for the manga.)
Okay, here are my favourite songs!
-
Jinzou Enemy: For some of the songs in the series, I think the vocaloid gets sort of drowned out by the sound of the instruments, so human covers sound better. But for this song, the very digital-sounding quality to the original Hatsune Miku vocals sounds really good. It took a while for this song to grow on me, but after several listens, it got really catchy. Fun fact: This was the first song that Jin released as a music producer, and that's why it doesn't have an official music video (the video I linked is a fan PV). Sidu first listened to this song on Daily Vocaloid Ranking, and then tweeted "Jin's song is really cool!" and then got insta-followed by Jin because at the time Jin only had like 80 followers. And then later on, they ended up working together on the KagePro song series, with Sidu doing the animations for many of the KagePro music videos.
-
Children Record - Nico Nico Chorus cover: Many consider this to be the "opening" for the song series. Nice and upbeat. I think a group cover is a really good fit for this song.
-
Kagerou Daze - Wannyanpuu PV: The classic time loop story. This song is basically the most popular song in the series. The linked video is actually a fan PV by Wannyanpuu; the official PV was animated by Sidu. But Wannyanpuu's fan PV became really popular, and Jin really liked it, so he requested for her to join his team. And so a number of the subequent KagePro music videos were animated by Wannyanpuu.
-
Otsukimi Recital - Yuikonnu cover: Immediately catchy. This utaite just sounds very Momo to me.
-
Yuukei Yesterday - LiSA cover: Happy peppy sounding song. I think LiSA's voice fits very well here.
-
Toumei Answer: Oof, this song. It hurts ;_; Also, I normally like utaite covers better than the original vocaloid, but IA actually sounds very good in this song, so I prefer this original version.
-
Ayano's Theory of Happiness - MARiA cover: I love this cover. The song is happy, and then sad
