The fastest way to get just about anywhere in Japan is by bullet train or what the Japanese call Shinkansen. There are no direct Shinkansens from Tokyo to Shimoda but it is possible to take a shinkansen half way, from Tokyo Station to Atami, and then transfer to a Super Express called Odoriko. The journey takes a mere 2 hours but will cost you ¥7090 one way.
You can save more than 50% by sticking to regular trains. You can read about the cheapest option こちら. There easiest way to get to Shimoda is to take 1 train direct from Tokyo to Izukyu Shimoda, find out more こちら, or call the JR East Info line for help:
(If you are calling from overseas, drop the first zero and add the country code 81)
***Please note, the name of the station you want to get off at is Izukyu Shimoda Station. If you use Hyperderia or Jorudan to search for train schedules and fares, you must use “Izukyu Shimoda” as your destination point. There are other towns and stations called Shimoda throughout Japan.
****One more note, when you are transfering in Ito or Atami, take care NOT to board a train bound for “Shimada” (with 2 A’s), be sure to look for a train bound for Izukyu Shimoda.