You can use Tuk Tuk (Tuk Tuk is typical Thailand rickshaw), Taxi or Chao Phraya Express Boat Service.
You can go by foot from Khao San road to Santichai Prakan Public Park (Suan Santichai Prakan) - it took only a few minutes. There is Phra Athit Express Boat Pier (Pier number 13). Wat Phra Kaew (King palace) is in the stop Tha chang (Pier number 9).
There is not necessary bought your own long pants (You need long pants, if you want to go to King palace). You can rent this pants for 1000 THB deposit inside Wat Phra Kaew.