Learn Italian: Fai presto e preparati per questa festa! - Hurry up and get ready for this party!

You are right that presto often means 'early' or 'soon'. However, the phrase fare presto is a very common expression that translates to 'to be quick' or 'to hurry up'.

Why is the word ti attached to the end of preparati?

Preparati is an informal command made from the reflexive verb prepararsi (to get oneself ready). In Italian, when giving an affirmative command to someone you address as tu, reflexive pronouns attach directly to the end of the conjugated verb.

Is fai the normal command form of fare? I sometimes see fa'.

Both fai and fa' (with an apostrophe) are completely correct and commonly used for the informal tu command of fare. You will hear and see both used interchangeably.