The film opens with the sound effects of Godzilla's stomping feet and culminating with his iconic roar. It was created by rubbing a leather glove on the strings of a base and then adding reverb.
The fishing boat scene is a direct representation of events that happened to the Fukuryu Maru (Lucky Dragon) earlier in 1954. You can read about it in the resources below.
Aftwards one survivor washes ashore at his home Odo Island.
Because of the loss of the ship and a lack of fish older people in the village blame an ancient mythical creature, Godzilla. That evening they perform a ritual dance and a village elder laments that they used to sacrifice virgins.
A typhoon ravages the village and the survivor of the fishing boat is killed. The implication is that you can't escape Godzilla just like you can't escape radiation or in the folk tradition an avenging spirit.
The final scene of the typhoon sequence is the destruction of the helicopter which foreshadows Godzilla's eventual attack on Tokyo.
After hearings in which Dr. Yamane, a paleontologist, discusses Yetis and sealed underwater caverns which may contain ancient life, it is decided that there should be a scientific expedition to Odo Island. The party includes Dr. Yamane, his daughter Emiko, and her boyfriend Ogata. They wave goodbye to the eye-patch wearing Dr. Serizawa, who we find out is Emiko's fiance.
On the island the scientist use a Geiger counter to discover that the giant footprints are radioactive and Dr. Yamane finds a living trilobite. The monster makes his first appearance at approximately 22:00 minutes into the film.
As the creature leaves we are shown his trail into the sea. Interestingly, Godzilla drags his tail, which is not how dinosaurs walk. They actually walk like birds with their tails in the air.