Best and most convenient way to go to Izu Kogen from Shimoda?
Izu Kyuko Line train is pretty much the only way to get to Izu-Kogen from Shimoda. There might be a bus but I don't think so.
Shimoda and Ito are different cities that is over 30km apart. Therefore, naturally I don't expect a bus route there.
Is it possible to do a day trip to izu kogen from Shimoda?
Yes, they're less than an hour apart by train.
You'd just take the Izukyu train. The normal commuter train will do. There are no buses.