Graded reader is レベル別読物 but I've only ever heard it used in terms of be foreign language reading materials. For materials for a first grader try searching for 一年生　読物 (first grade reading material) or 二年生　読物 for sexond grade materials. Here is a collection of short passages for first graders so you can see what I mean. https://www.amazon.co.jp/10分で読めるお話-一年生-岡-信子/dp/4052022033 To find a list of stories recommended for various grade levels search for 一年生　推薦文庫. It sounds like you want to use materials for native speakers of Japanese but I would also highly recommend the よむよむ文庫 graded readers. They were written for people learning Japanese as a foreign language and they are very well done. You can get them with CDs so you can listen as you read along. Definitely consider these as well. https://www.amazon.com/Japanese-Graded-Readers-Level-Audio/dp/4872176243/ref=pd_aw_sbs_14_3?_encoding=UTF8&pd_rd_i=4872176243&pd_rd_r=fdf63d44-001f-11e9-bcdd-7501865a8e6d&pd_rd_w=DzjVY&pd_rd_wg=4xzRg&pf_rd_p=926ebe02-3236-40c6-ac63-01ad178f498a&pf_rd_r=C7MMEAKAEKMBAKTRK053&psc=1&refRID=C7MMEAKAEKMBAKTRK053 https://www.amazon.com/Japanese-Graded-Readers-Level-Audio/dp/4872177118/ref=mp_s_a_1_1?ie=UTF8&qid=1544847153&sr=8-1&pi=AC_SX236_SY340_FMwebp_QL65&keywords=japanese+graded+readers+level+0

By way of introduction, we are a bilingual Japanese/American family living in Japan. A children's picture dictionary is ejiten (絵辞典）or kodomo zukan (子供図鑑）. The Junko Yoshida kodomo zukan that PP linked to are really popular but you are right, they are board books meant for younger children (0-4 years). Each one covers specific topics and I think they have a fair amount of vocabulary for those topics. You would just need to get more than one for sure to be sure to cover the topics you want to look at. For example, we have one that covers "around town" (shops, hospital, train station, festivals, etc.) and a general one that covers animals, fruit/veg, vehicles, insects, flowers, body parts, numbers. I'm not sure I would pay 18 USD for them though (the retail price in Japan is 1000 yen but you can easily get them used for around 500) but otherwise they might actually serve you well, if you got most/all of the series (I think there are 4 or 5 of them) . I think this one might be more along the lines of what you are looking for though https://www.kumonshuppan.com/gakusan/gakusan-syousai-jiten/?code=61023 If you click below the picture of the cover where it says 立ち読みする in blue you can see some sample pages (needs flash). It's also on Amazon but with no sample pages. https://www.amazon.com/Kumon-no-kotoba-ejiten-Tooru/dp/4774313386/ref=sr_1_1?s=books&ie=UTF8&qid=1544533744&sr=1-1&keywords=kumon+ejiten Here's another that might be what you are looking for https://www.amazon.co.jp/学研-ことば-えじてん-無藤隆/dp/4053032946/ref=sr_1_1?s=books&ie=UTF8&qid=1544534199&sr=1-1&keywords=学研絵辞典 The second one has twice as many words (around 3000) as the first one (around 1500). That's just based of the publisher's specs online. I haven't seen either IRL. These are both geared more towards younger kids (3-6) but might work for your boys? This one is for younger kids too though (2-6 is the age range the publisher gives).

Not "learning" apps per se but Skybrary saved my life when I was pregnant and sick and then when we had a newborn. In the same vein- story podcasts like story time and circle round are awesome when you don't have the energy to read aloud.

We just finished Foundations A with our 4.5 year old and have started on B. We love it. I love that it is open and go. My only prep is to get out the materials for the lesson and so a 30 second scan of the lesson before I teach it. DD loves that it is fun. Some of the game I initially thought sounded a bit dumb but then those are invariably the ones my DD lives the most lol. Lessons have a variety of activities with plenty of opportunities to move so these are not long stretches of seat work. DD doesn't beg for "Dragon School" every day like she does Key to learning @ Home activities (that program is brilliant) but she's happy to do it when I bring out the books. If you decide to give it a try I would give it at least 7or 8 lessons before you decide if it is for you. Also watch all the teacher training stuff online- very helpful! The cursive first is not a hill to die on IMO. I understand the logic and we thought about going that direction at first but we went with manuscript for various reasons. So you could keep doing manuscript or switch to cursive. I doubt your son would find it confusing.