The cheapest way to get to Shimoda is to take the Odakyu Line train from Shinjuku to Odawara and transfer to the Tokaido line bound for Atami or Ito, then finally switching to Tokyu's Izu Kyuko line from there. The train from Shinjuku to Odawara takes 93 minutes so be sure to catch an express. The entire journey takes approximatly 3hrs and 40 minutes and costs only ¥3070.
I suggest you use Hyperdia to check for train schedules. Click on "search details" and make sure you uncheck "Bullet Train", "Nozomi", and "Limited Express Trains" to find the cheapest routes.
You can also ask the super friendly folks at the JR East English 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 Hyperdia 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.