Disclaimer: I'm by no means an expert in the Shinto religion, I will list sources at the end of each section.
Difference Between Shrines and Temples
It's important to first know the difference between Shrines and Temples. Shrines are places of prayer for the Shinto religion while Temples are only associated with the Buddhist religion. It can sometimes be difficult to tell the two apart but there are some characteristics that distinguish the two.
1. Always have a red torii gate
2. Have guardian dogs, lions or fox statues that sit on each side of the entrance outside a Shinto shrine
3. Near the Shinto shrine is a purification fountain to wash your hands and face
1. Always have an image of Buddha inside
2. Have a large incense burner at the front of the temple
3. Have a pagoda near the temple
What Are Shrines? What Do You Do Once You Get There?
Shrines are specific to different kami and are a place of worship and prayer for the Japanese people. To enter a shrine you go through a red gate called torii. Once you enter through the gate you will find a water fountain with a bamboo ladle. Before you enter the shrine you must wash your hands and mouth to purify yourself.
Once you enter the shrine there should be a bell with a rope hanging from it. Ring the bell to summon, toss 5 yen coins into the slates before the alter for good luck and clap your hands three times to summon the kami. Put your hands together to to pray if you wish.
Some Notable Kami
Amaterasu-O-Mi-Kami: Amaterasu is the most famous kami. She is the sun Goddess, and is the most important kami in the Shinto religion. Amaterasu is also the ruler of Takama no Haru or the High Celestial Plain, this is the domain of kami. The Ise Grand Shrine is dedicated to Amaterasu and is the most sacred shrine in the Shinto religion
Raijin & Fujin: These two Shinto kami commonly appear together. Together they are the Gods of thunder, lightening and storms. These deities are feared by many because of the damage they cause during storms. These kami sometimes appear at the entrance of shrines and act as protectors. Raijin is the God of thunder, lightening and storms while Fujin is the God of wind and is depicted holding a bag of wind.
Inari Okami: Inari is another important kami and is responsible for everything that is essential to Japanese life including rice, fertility, sake and success. Inari uses foxes to send messages to earth, this is why foxes are so revered in Japanese society. Often, you'll see tiny shrines dedicated to foxes and fox statues near Japanese shrines.
Izanami & Izanagi: These are the kami of creation. Both kami created earth by stirring the sea between heaven and earth with a jewel studded spear.
These are just some kami, obviously I can't list them all. Shrines are so magnificent and beautiful, you have to visit at least one when you travel to Japan. I hope this article gave you some ideas about which shrines to visit and how to properly respect them. If you aren't religious or don't wish to acknowledge other Gods in fear of violating your religion than you don't need to pray at shrines, just admire the architecture. Happy traveling :)