You want "first class" because you want some extra comfort and luxury. You probably know that intercontinental flights might offer first class, business class, and economy cabins, but that most carriers don't offer intercontinental first class.

JR has analogous seating in separate cars: Gran Class, Green Car, and Ordinary seats.

Your trip to Kanazawa is on the Hokuriku Shinkansen line. The trains on this line offer Gran Class and Green Car premium seating. You should prefer the Kagayaki train. Buy this from JR East.

Buy the rest of your tickets from JR West.

From Kanazawa to Kyoto you would take the Hokuriku Shinkansen, again in Gran Class or Green Car from Kanazawa to Tsuruga where you will transfer to a limited express train to get to Kyoto. That train, like the rest of the trains on your trip, does not offer Gran Class.

Kyoto- Hiroshima- Okayama is all on the Sanyo Shinkansen line. Take a Nozomi train to Hiroshima to have Green Car. For Hiroshima- Okayama it used to be that some shinkansen did not offer Green Car. This may have changed. In any case, this is only a ~40-minute ride so an ordinary seat will do. You can save 4 minutes and get a Green Car if you take a Nozomi train.