Religion - Two major religions of Japan are Buddhism and Shintoism
If you are interested in Japanese religious concepts, it is better to visit both Buddhist temples and Shinto shrines. Sensoji temple in Asakusa, the oldest temple in Tokyo, is always crowded with tourists from overseas because of its famous traditional shopping arcade (Nakamise), but most of them do not pay much attention to the shrine sitting next to the temple, which shows the syncretism of Shinto and Buddhism. It is one of the keys to understand Japanese religion correctly.
Talking about a shrine worth visiting, very famous Meiji shrine located near Shibuya is raised. It is a typical Shinto shrine and you can realize the differece between Shinto shrines from Buddhism temples. Being located just next to famous fashion town Harajuku, Meiji shrine is always crowded with many tourists from overseas.
If you have come to Tokyo before, you might have visited above two places. In such case, I propose you visit Kanda Myojin, which I think is the richest shrine in Japan, located near Akihabara. Because a deity of the shrine is for commerce and business, a lot of Japanese companies in Tokyo make a donation to Kanda Myojin. It opened new communication hall for visitors at the end of last year, where you can buy many interesting Shinto goods or souveniors.
Another idea is to visit shrines located at the top of mountains if you are much interested in the Japanese religion and have enough time and physical strength. Mt.Takao and Mt.Mitake, both of which are located at about 1-1.5 hour trip from central Tokyo, are known as their long history of mountain asceticism. Don't you like mountain climbing? Don't worry, You can ride cable railways to go up the mountain. You can enjoy both nature and culture at a same time!
In particular, one of my friend guides is a professional guide for Mt. Mitake. He plans one day tour to Mt. Mitake and Mitake shrine including Shinto training (waterfall purification, mountain traverse, and meditation). If you are interested in these experiences, I will pass your inquiry to my friend guide (his tradename is Taiyaki Guide).