Shinto beliefs can vary and, lacking a dogma or a pope figure, often vary by individual, and even by shrine. To give a better idea though, I will give you some broader things most people believe: there are many kamisama, they can give blessings or harm, and there are rules that should be followed. The exact nature of the rules can vary by shrine, but some universal ones are things like handling sacred items with clean hands and care, not opening omamori, and so on.
One of the most frequently asked questions I get about Shinto is how to make offerings, when, etc. This is natural, as giving offerings is the most important core part of Shinto practice. The standard idea is to offer rice, water, and salt every day. Offerings get removed and consumed, ideally by humans, but watering plants with the offered water is okay too.
See my kamidana page for information on altar formats.
First, have clean hands, gargle and be dressed. The general rule is to bow twice, clap twice and bow again, but this may vary by shrine or by sect/faith. You can pray or wish in your heart before or after you clap twice. You can also chant norito, which are sacred prayers. If this is not always possible, my experience as well as advice I have received is that kamisama will understand.
Make sure to bow before you take down offerings as well.
Buddhism and Shinto easily coexist, and have historically mixed a lot. Sometimes the way they mixed became a power struggle, with one side seeking to be the 'source' of the other. On the other hand, the Japanese Imperial Government forcibly separated the two, doing irreperable damage to many shrines and temples nationwide.
Many folk traditions, shrines, and temples have managed to keep syncretism alive, however.