I'm not at all an expert on Shinto shrines, but here are a few of the ones I thought were worth seeing (I also recommend the Miyajima shrine mentioned earlier):

Fushimi Inari Taisha in Kyoto - has a path that is lined with hundreds (thousands) of torii, like going through a tunnel of torii. Makes for good photos.

Toshogu in Nikko - Tokugawa Ieyasu is buried here. My understanding is that this is not a very traditional Japanese shrine, it has heavy Chinese influences. It's incredibly gaudy compared to most shrines, with gold leaf, detailed engraving, and vibrant colors everywhere.

I'm not sure if you have a specific interest in Shinto shrines, but the other major religion in Japan is Buddhism, and there are also many Buddhist temples that are interesting to see - off the top of my head I would say Kinkakuji and Kiyomizudera in Kyoto and Todaiji in Nara.