Does gargling or rinsing my mouth break the fast?

What breaks a fast is something deliberately reaching the stomach. Rinsing is not that, and so it does not invalidate it.

The boundary of the fast is set in the same verse that sets its hours: “It has been made permissible for you the night preceding fasting to go to your wives for sexual relations. They are a clothing for you and you are a clothing for them. Allah knows that you used to deceive yourselves, so He accepted your repentance and forgave you. So now, have relations with them and seek that which Allah has decreed for you i.e., offspring. And eat and drink until the white thread of dawn becomes distinct to you from the black thread of night. Then complete the fast until the night i.e., sunset. And do not have relations with them as long as you are staying for worship in the mosques. These are the limits set by Allah, so do not approach them. Thus does Allah make clear His verses i.e., ordinances to the people that they may become righteous.” (Qur’an 2:187)

In practice:

• Rinsing the mouth and drawing water up the nose are part of wudū’, which continues five times a day through Ramadan. Nobody drops wudū’ because they are fasting.
• The only advice attached is to be moderate with the nose, to reduce the chance of water going down unintended.
• Brushing your teeth does not break the fast either. On toothpaste, some prefer to avoid it out of caution because of the taste, but that is not the same as saying it invalidates.
• If something goes down unintentionally, there is nothing to make up: what is involuntary does not count.

A useful note for a first Ramadan: nearly all of these worries share one answer, and it comes from one idea. A fast is broken by what you do on purpose, not by what happens to you. If you did not intend it, and it was not food, drink or intimacy, you are almost certainly fine.