After all of this time, I've failed to say where I've been reading Apocalypse no Toride. I’ve used several sources. Since it’s only available in a magazine in Japan, I (in the USA) can only read it as a scanlation from online sources. Manga that have been scanned and then translated into another language other than its original language is called a scanlation (scan + translation).
This is a site to read manga for free. It hosts lots of English translated manga. This is the first place I discovered ApnoTo early this year. As of today (12-12-13) they have chapter 16 posted.
This is another free manga site. They too have English scanlations. As of today (12-12-13) they have chapter 20 posted.
This anime and live-action streaming service has just begun to host manga, with Apocalypse no Toride being one of the first manga to be hosted. You must pay for this service in order to get full access to anime, live action dramas, and manga. Go here to compare plans. I’ve had the premium plain ($11.95 a month) for over a year, so I cannot say what the Free or Anime memberships will allow you to watch or read. As of today (12-12-13) they have chapter 27. This site is ideal for reading ApnoTo because they have simultaneous publishing, meaning that the day the manga is published in Japan, it will be available on the Crunchyroll site. This site is also ideal because there are no watermarks on any of their images. So now, when I find a must-have image, I can get it from there (by print-screening people, you cannot simply [as far as I know] right-click and save an image from this site.)
There are probably other sites that have ApnoTo scanlations, some in languages other than English. But those are the main three I’ve read it on.
One final note. Both Spectrum Nexus and MangaReader use ApnoTo scanlations from Scans Vendetta Scans.Seems like this scanlation team may have cornered the market on this manga :)