It is called takkyubin service and is the standard way most Japanese travelers deal with large luggage. They send it ahead and travel with only a light overnight bag. Yamato is the largest but there are several companies and all are reliable. You can send the bag ahead several days if needed and the hotel at the other end will hold it for your arrival. The front desk at almost any hotel can handle the details for you. You could even send bags ahead from NRT to Kyoto as there are takkyubin counters in the arrivals area of the airport. That might not work the best in your case as you are going directly to Kyoto and the bags would take a day to catch up with you.
I am sure you realize this but just to be sure - you can't take a train "from NRT to Kyoto." You must take an airport train from NRT into Tokyo and then board the shinkansen (bullet train) to Kyoto or Osaka. For a direct move from NRT to Kyoto, the better choice of transfer station is Shinagawa Station, not Tokyo Station. The process would be to pass through Immigration, pick up luggage, pass through Customs, exchange money or visit an ATM for cash, visit the basement floor Japan Rail Service Center to exchange your Rail Pass and make seat reservations for both trains, take the Narita Express train to Shinagawa (about 1 hour,) transfer to the Kyoto-bound train (about 2-1/2 hours,) arrive Kyoto Station.
Since you will be using the Narita Express and doing the round trip to/from Kyoto-Osaka, the Japan Rail 7-day ordinary Pass will save you money. You must purchase the Rail Pass before you arrive in Japan and exchange the paper order for the actual Rail Pass after you get there. With the Rail Pass, all long distance travel as well as the trip from Kyoto to Osaka (about 30-45 minutes) and any day trips, such as to Nara, etc., are covered for free on JR trains. For local trains, you do not even need tickets. Just walk on the desired train and show your Rail Pass. If you elect not to purchase a Rail Pass, you just buy tickets for everything as you go.