- Kita-Kamakura is different from Kamakura?
"KITA" means "North". "KITA-KAMAKURA" means "North part of KAMAKURA". How do you find "KITA-KAMAKURA" instead of "KAMAKURA"?
- If going from Hakone to Kita-Kamakura (or Kamakura?), I should take JR?
You need to change train ODA-KYU to JR at "小田原:ODAWARA" and "大船" ( sorry, I did not mark them on the map). it will take 80 to 100 minutes.
- If going from Hakone to Enoshima, I should take the Romancecar?
Yahoo transit recommends me to change train also at 小田原:ODAWARA and 藤沢：FUJISAWA while I thought changing train at SAGAMI-OHNO is better (but I'm not sure if RomanceCar stops at SAGAMI-OHNO).
>Am I correct I should take Enoshima Railway from Sagami-Ohno (a place in Hakone?)?
As I prefer to the ENOSHIMA-line of ODAKYU, I told you this route. You can take either ODAWARA route or SAGAMI-OHNO route. (And it needs only one train change)
>how long does it take from Hakone to either Kita-Kamakura or Enoshima?
It will take 80 to 90 minutes from Hakone to Enoshima (ODAWARA route).
Here in this video, they introduce you to Hakone. It is basically in Japanese but interviews to foreigners are in English.
Edited: Because I was wrong misunderstanding Enoshima-Line of ODAKYU with ENODEN (Enoshima Dentetsu). ENODEN is small local line near Enoshima and can not reach to SAGAMI-OHNO.