I think your travel plan is doable and quite normal for first time traveller to Japan. I would suggest a slight change.
Hiroshima 3 nights (Himeji as a stopover)
Kanazawa 2 nights
Tokyo 4 nights (Kamakura day trip)
Kyoto 5 nights (Osaka day trip if needed)
Your travel dates will be so-called gTsuyu seasonh which is hot and rainy days. This is not continuous rain but you should expect half of your days will be rainy. This is not so bad because if itfs sunny it will be extremely hot in this season. Another bright side is the hotels tend to be vacant so you can choose from many. Tourist spots will not be overcrowded.
All of your destinations are cities so you donft need to bring many belongings: if you are not picky persons, your accommodations will provide towels, nightwear, toiletries, etc. Even if you want something you can easily find stores selling them. Note that Japan is extremely cheap these days.
I recommend pack light. 30-40 L backpack for each person is enough. Bring 3 days of clothes and wash them in your hotel. If you choose apartment hotels they should have in-room washing machine. If you still want to bring big suitcase, consider delivery system.
https://www.japan-guide.com/e/e2278.htmlAs for travel passes, your first 4 days will be covered by JR West Kansai Hiroshima Area pass.
https://www.westjr.co.jp/global/en/ticket/pass/kansai_hiroshima/On day 4 you will go from Hiroshima to Kanazawa. Hiroshima to Tsuruga is covered by the pass (you should transfer at Shin-Osaka, not Kyoto since Shinkansen between these two cities is not covered by the said pass). So you should buy somewhat tricky base fare ticket :Tsuruga to Kanazawa to Tokyo to Kyoto. This will save some money than to purchase each leg (Tsuruga to Kanazawa, Kanazawa to Tokyo and Tokyo to Kyoto). You can buy JR tickets in almost all major JR train stations (in your case, Kansai Airport, Himeji, Hiroshima). Note that you need separate Limited express ticket.
https://www.japan-guide.com/e/e2359_001.htmlYou can be flexible for your trip. For example if it's heavily raining you can skip Himeji castle, etc.