Shizuoka Airport is less crowded and complicated than Narita if you go to Atami. First take the bus to Shizuoka Station (see From Shizuoka Airport to Shizuoka) and take the local train bound for Atami. The train to Atami runs about three times an hour in average. You can get there from Shizuoka station using Shinkansen Kodama Express which is a lot faster but a lot more costly. Haneda is often picked because of frequency of flights and better accessivity. But one thing that Train ride from Shizuoka beats the other means is the view of Mount Fuji up close.
To get to Hamamatsu, Kanaya, Kakegawa, Fukuroi, Iwata from Shizuoka Airport, take the bus bound for Shimada station and then take the train bound for Hamamatsu, Toyohashi, and Ogaki. Although the bus service is more frequent than to the Shizuoka station, these lines often use a demand taxi for 8 people so be sure to make a reservation for seat through your travel agency. You can make a reservation up to one day before you take the bus. If there are vacant seats you may take the bus(or taxi) without reservation. The bus(taxi) starts from #4 bus gate. Bus fare is 500 yen for adult. The bus line might be cancelled if there are no reservations. If you are staying at the hotel in Kakegawa or Shimada ask them if there can pick you up at the airport or reserve the bus for you.
If you arrive at Shizuoka Airport from Seoul, Taipei, or Shanghai and are new to Japan, I do recommend that you take the bus to Shizuoka station. Shizuoka station has two information offices and they are really helpful to lead you to the destination you want to go. The bus leaves only when there are incoming flights. So you may not have enough time to shop around at the airport. (Shizuoka has a lot more places to shop and dine).
When you come out from the baggage claim area you will find a small shop, an information office and Rodin's The Thinker. To get the bus line turn left at the Thinker sculpture to get out of the Terminal Building and walk leftward. You will find bus stops. To get to Shizuoka station take the bus departing from #5 gate. The bus charge is 1,000 yen for adult.