The temple mount is the holiest Jewish site, but also the third holiest Muslim site. It is also a geopolitically sensitive site, probably the most sensitive in the country and various rules and limitations apply. Please note that these can change daily!
Dress code: Modest dress is required, trousers/pants/skirts must cover your knees, closed necklines, and sleeves are required. Occasionally, there is a request to cover your hair as well.
Religious artifacts: You are not allowed to bring any religious books, artifacts, symbols, etc, with you.
Identification: You must bring your passport or Israeli identity card if you are an Israeli citizen
Food & Drinks: During the month of Ramadan, you will not be permitted to enter with food or drinks.
Prayer: all prayer, except Muslim prayer, is forbidden on the site
Religious Jews: Religious Jews are required to enter and remain in a large, guided group, with police escort. This will also ensure that you do not go beyond the areas accepted by the rabbis who permit visiting the temple mount. If you are not interested in being a part of this group, don't dress like a religious Jew.
You will get much more out of this visit if you go with a guide!