I just did this trip last Sat
I went from Shinagawa to Kyoto on the Noizumi (sp?) trains.
It cost me about 27,000K yen RT total. That was for a non-reserved ticket. I also found it hard to figure out how much the ticket was and I ended up buying the ticket from the ticket window in Shinagawa (as opposed to the automatic ticket machine). The automatic ticket machine asked me for a 4 digit PIN # (even when I tried to use my credit card). It wouldn't accept the PIN # for my debit card thus you should make sure you either have the cash or just go to the ticket window to buy the ticket. The non-reserved seats are slightly cheaper than the reserved seats - on the Noizumi there were three cars for the non-reserved seats (cars 1 - 3, with car 3 being the smoking car and 1-2 non-smoking). I had to buy 4 tickets total - it's a bit complicated which is another reason to go to the train ticket office in Narita as opposed to the ticket machine. The people at the ticket windows speak some English and the Shinkansen trains have really clear announcements in English. The electronic display boards also switch between Japanese and English which helped me out when I tried to find my train.
One warning though, while the Narita Express trains have luggage racks at the ends of the cars, there were no such racks on the Shinkansen train from Shinagawa to Kyoto.
I also took the Narita Express from Narita to Shinagawa, except that I got off at Shinagawa and stayed at a hotel for 5 nights before making my weekend trip to Kyoto. Since you don't have that requirement you can do as the other posters suggest and just change trains at Tokyo Station.
Also, when I caught the Shinkansen, the destination displayed was 'Shin-Osaka' although every once in a while the board would change and display 'stopping in Nagoya and Kyoto'. I'm not sure what destination the non-Noizumi trains would display, but be prepared for the boards not to display 'Kyoto' as the final destination.