There is no luggage check-in on Shinkansen. You take your bags with you and there are no fees. However, since October 2020 JR has introduced a rule for large luggage: bags exceeding 160 cm total dimensions require a "large baggage reservation" (予約制大型荷物) — book this when buying your ticket. There is no extra charge, just a designated storage space behind the last row of seats.
Overhead racks hold smaller bags and backpacks. Large suitcases go in the dedicated luggage space at the end of each carriage. On the Nozomi, Hikari and Kodama services there are also lockable compartments behind the rear seats for reserved large-luggage spots. Luggage is not supervised — keep valuables with you.
For very large items such as bicycles or surfboards, contact JR in advance. Most stations have coin lockers in a wide range of sizes — extremely useful if you're spending the day sightseeing before or after your train and don't want to drag a suitcase around Kyoto or Tokyo.