Hello. I don't know if you still need an answer of how to go to Shibazakura. But actually I went yesterday (helping me with the answers on this page...) So I think I can help you back with maybe some clearer information. I finally took the bus because it seemed more comfortable (at least you are seated the whole time!)
1) Go to Shinjuku station and try to find the west exit. From here, it is really confusing because there are a big bus stop place when you go to street directly from the west exit. But it's not here! But if you exit there, just take to the right till the end of the main bus stop place and on the corner you should see a building "Bus Keio Line" it's here (there is a big stop number 50 there and a huge camera shop) If you are lucky maybe you will find the signs "express highway bus" on the underground passage but personally I found it after 30 minutes of desperate research!!! I didn't book my bus, but could buy one ticket, nevertheless I had to wait 1 hour for a bus with vacant seats. It cost 1700 yen one way and take about 2 hours to go to Kawaguchiko.
2) From Kawaguchiko station you will see a lot of pink flags, near the gateway 7. It's here you can buy a ticket (1800 yens) for the package "shuttle bus to go and return and ticket for entrance in the park". The journey is about 30 minutes.
3) Enjoy Shibazakura! That's wonderful!!! And on a clear day you have a wonderful view on Mount Fuji! I had the chance to be there on a clear day! That's incredible!!
4) Be careful of timetable when you go back, for the shuttle bus (we waited 40 minutes!) and for the main bus back to Shinjuku if you still don't have your ticket. (We wanted to walk around Kawaguchiko, but when we asked when we could go back they told us that the only buy with vacant seats was 30 minutes after...) Last bus leaves at 20.10 but we had to go back at 17.40... So be carefull!! (Or book before ^^)
You should just know that this year all the flowers blossomed really early and even if yesterday it was still beautiful, we could see already some big patches of green grass... So maybe in a couple of days it'll be all gone... But there are many other cool stuff to do (cruise on Kawaguchiko lake or rope way to see some wonderful landscapes... (both 10 minutes walking from the station) and you can ask direction in the tourist office there, and they speak English!)
I hope I answered to all your questions =) And that it's going to help you! On
http://www.shibazakura.jp you can have a look on the pictures of the site they update everyday to have a look on what the site still look like at that precise day! And don't forget, it you are lucky, on a clear day, mount Fuji is amazingly beautiful!! =)
Enjoy your stay in Japan!!