There are many acts of worship which are prescribed for the Muslim to do on this day. These include:

1 - Salaat al-Jumu'ah (Friday prayer)

Allaah says:

"O you who believe (Muslims)! When the call is proclaimed for the Salaah (prayer) on Friday (Jumu'ah prayer), come to the remembrance of Allaah [Jumu'ah religious talk (Khutbah) and Salaah (prayer)] and leave off business (and every other thing). That is better for you if you did but know!"
[al-Jumu'ah 62:9]

Ibn al-Qayyim (may Allaah have mercy on him) said in Zaad al-Ma'aad (1/376): Friday prayer is one of the most important obligations in Islam, and one of the greatest gatherings of the Muslims. Gathering on Friday is more important and more obligatory than any other gathering apart from 'Arafah. Whoever neglects it, Allaah will place a seal on his heart.

It was narrated from Abu'l-Ja'd al-Damari (radiallahu anh) - who was a companion of the Prophet (salallahu 'alayhi wa sallam) - that the Prophet (salallahu 'alayhi wa sallam) said: "Whoever neglects three Jumu'ahs, Allaah will place a seal on his heart." [Narrated by Abu Dawood, 1052]

It was narrated from 'Abd-Allaah ibn 'Umar and Abu Hurayrah (radiallahu anhum) that they heard the Messenger of Allaah (salallahu 'alayhi wa sallam) say on the minbar: "Let people stop neglecting Jumu'ah, or Allaah will place a seal on their hearts, then they will be among those who are neglectful." [Narrated by Muslim, 865]

2 - Making a lot of du'aa'

On this day there is a time when a person's Lord will answer his du'aa' - by Allaah's leave.

It was narrated from Abu Hurayrah (radiallahu anh) that the Messenger of Allaah (salallahu 'alayhi wa sallam) mentioned Friday and said: "On this day there is a time when no Muslim stands and prays, asking Allaah for something, but Allaah will grant him it" - and he gestured with his hands to indicate how short that time is.
[Narrated by al-Bukhaari, 893; Muslim, 852]

Also see: Time of the acceptance of Dua - By Shaykh Riyad ul Haq (It is explained that one of the most famous views (and this is the opinion of the majority of the scholars), is that the time of the acceptance of Du'a is after Asr prayer until sunset, on Jumu'ah (friday). And most preferably right towards the end. Imam Ahmed (rahmatullahi alaih) says that most of the ahadeeth support this)

3 - Reading Soorat al-Kahf.

It was narrated from Abu Sa'eed al-Khduri (radiallahu anh) that the Prophet (salallahu 'alayhi wa sallam) said: "Whoever reads Soorat al-Kahf on Friday, he will be illuminated with light between the two Fridays." [
[Narrated by al-Haakim]

Also see: Virtues of Surah Kahf - By Shaykh Riyad ul Haq, and The Virtues of Reciting Surah Kahf on Jumu'ah (Friday)

4 - Sending a lot of blessings upon the Prophet (salallahu 'alayhi wa sallam)

It was narrated from Aws ibn Aws that the Prophet (salallahu 'alayhi wa sallam) said: "The best of your days is Friday. On that day Adam (peace be upon him) was created; on that day he died; on that day the Trumpet will be blown; and on that day all of creation will swoon. So send a great deal of blessings upon me, for your blessings will be shown to me." They said, "O Messenger of Allaah, how will our blessings upon you be shown to you when you have turned to dust?" He said, "Allaah has forbidden the earth to consume the bodies of the Prophets, peace be upon them."
[Narrated by Abu Dawood, 1047; classed as saheeh by Ibn al-Qayyim in his comments on Sunan Abi Dawood, 4/273]

He said in 'Awn al-Ma'bood:
Friday is singled out because it is the best of days and the chosen day. So there is an advantage to sending blessings upon him on that day that is not present on any other day.

Along with these virtues and acts of worship, the Prophet (salallahu 'alayhi wa sallam) forbade singling out Friday for fasting, or singling out the night of Friday for doing acts of worship that are not narrated in sharee'ah.

It was narrated from Abu Hurayrah (radiallahu anh) that the Prophet (salallahu 'alayhi wa sallam) said: "Do not single out the night of Friday for qiyaam al-layl, and do not single out the day of Friday for fasting, unless that is part of the habitual fasting of any one of you."
[Narrated by Muslim, 1144]

Al-Nawawi (rahmatullahi alaih) said:
This hadeeth clearly states that it is forbidden to single out the night of Friday for prayer or the day of Friday for fasting. It is agreed (among the scholars) that this is makrooh (disliked).

He also said:
The scholars said that the reason why it is forbidden to single out Friday for fasting is that Friday is a day of du'aa', dhikr and worship, such as ghusl, going early to the prayer, waiting for the prayer, listening to the khutbah and reciting a lot of dhikr afterwards, because Allaah says: "Then when the (Jumu'ah) Salaah (prayer) is ended, you may disperse through the land, and seek the Bounty of Allaah (by working), and remember Allaah much"
[al-Jumu'ah 62:10]

And there are other acts of worship to be done on this day. So it is preferable not to fast then, as this will help a person to perform these duties in an energetic manner and enjoy them without feeling bored or tired. This is like the pilgrim on the day of 'Arafah in 'Arafah, because the Sunnah is for him not to fast on that day for the same reason... This is the reason for not allowing singling out Friday for fasting.

Also see: Virtues of Jumu'ah (Friday)