The Gurmantar or "Waheguru" is the most common way you will hear Sikhs describe the One. When you break down the word Waheguru, "Wah(e)" is like wow or wondrous and "Guru" means teacher (or someone who brings you from darkness to light). So the most simple translation of this word is "Wow God" while the deepest translation would be "Praise to the One who brings us from Darkness to Light". This is also the most common mantra used to do Simran (which is the remembrance of God).
The Mool Mantar
Next to the Gurmantar, the Mool Mantar is the second most common mantra Sikhs use to remember God. It goes...
Hai Bhee Sach
Nanak hosee bhee Sach
For a good understanding of what the Mool Mantar means, watch this playlist by Bhai Satpal Singh:
Nanak Naam (Mool Mantar Interpretation)
These are some of the characteristics of God laid out by Guru Nanak in the very beginning of the Shri Guru Granth Sahib. It is also the first part of Japji Sahib (one of the morning prayers Sikhs are supposed to recite). Now it should be noted that there is a little division between some Sikhs on where exactly the Mool Mantar ends. Some say it goes just to "Gur Prasad", while others say it ends with "Nanak hosee bhee Sach". This is a bunch of silliness and you should come to your own
The Ardas is the formal Sikh prayer that basically opens or closes any Sikh related activity. If there's something you should memorize, it definitely should be the Ardas. I think it is also important to know that the first part of the Ardas was authored by Guru Gobind Singh and is taken out of the Dasam Granth. The rest was added later on.
Here's a link to the full Ardas here:
Ardas (with English Translation)
Common Slogans you will hear Sikhs chant from the Ardas are....
"Degh Tegh Fateh!" - "May the Kitchen and Sword be Victorious"
"Nanak Naam Chardi Kala Tere Bhane Sarbat da Balla" - "Nanak, with Naam comes Chardi Kala and with your blessings, peace for everyone."
Anand Sahib is a collection of hymns in Sikhism, written in the Ramkali Raag by Guru Amar Das, the third Guru of the Sikhs. Anand means "happiness" or "bliss". This bani concludes the morning prayers of the Sikhs and also concludes all Gurdwara services (with the exceptions of ones that take place late in the afternoon).
"Waheguru ji ka Khalsa. Waheguru ji ki Fateh"
This is the traditional Sikh greeting passed down by the tenth Guru, Guru Gobind Singh ji. It means, "The Khalsa (or Pure Ones) belong to God! Victory belongs to God as well!" This is also called the "GurFateh".
"Bole so Nihaal! Sat Shri Akaal/Akaaluh!"
Though there can be different versions of the jakara, "Bole so Nihaal! Sat Shri Akaal" is the most common battle cry you will hear in the Sikh community today. It means "One will be blessed/fulfilled who proclaims that the Timeless One is the ultimate Truth". This is also said at the end of every Sikh activity.
I also think that it is important to note here that "Sat Shri Akaal" (or as some say, Sat Shri 'Kaal) is the traditional Punjabi greeting. Besides the GurFateh, this is the second most common greeting you will hear Sikhs say.
"Guru Maneyo Granth"
"Guru Maneyo Granth" (Granth Be Thy Guru) refers to the historic statement of the 10th Sikh Guru, Guru Gobind Singh (1666–1708), shortly before his demise, on affirming the sacred scripture Adi Granth as his successor, thus terminating the line of human Gurus. It is often quoted at the end of the Ardas.
It says: "Agya bhai Akal ki tabhi chalayo Panth Sabh Sikhan ko hukam hai Guru manyo Granth Guru Granth Ji manyo pargat Guran ki deh Jo Prabhu ko milbo chahe khoj shabad mein le Raj karega Khalsa aqi rahei na koe Khwar hoe sabh milange bache sharan jo hoe."
"Under orders of the Immortal Being, the Panth was created. All the Sikhs are enjoined to accept the Granth as their Guru. Consider the Guru Granth as embodiment of the Gurus. Those who want to meet God, can find Him in its hymns. The pure shall rule, and impure will be no more, Those separated will unite and all the devotees shall be saved."
Raj Karega Khalsa is a common slogan you will hear taken from this proclamation. It means "The Khalsa (Pure Ones) will rule!"
"Jithe Jaye Bahe Mera Satguru"
This shabad comes from Ang 450 in the Shri Guru Granth Sahib. This is often recited during Sukhasan (when the Guru is being retired for the day). It says...
jithai jai bahai meraa satiguroo so thaan suhaavaa raam raaje ||
Wherever my True Guru goes and sits, that place is beautiful, O Lord King.
gurasikhee(n) so thaan bhaaliaa lai dhur mukh laavaa ||
The Guru's Sikhs seek out that place; they take the dust and apply it to their faces.
gurasikhaa kee ghaal thai piee jin har naam dhiaavaa ||
The works of the Guru's Sikhs, who meditate on the Lord's Name, are approved.
jin(h) naanak satigur poojiaa tin har pooj karaavaa ||2||
Those who worship the True Guru, O Nanak - the Lord causes them to be worshipped in turn. ||2||
gurasikhaa man har preet hai har naam har teree raam raaje ||
The Guru's Sikh keeps the Love of the Lord, and the Name of the Lord, in his mind. He loves You, O Lord, O Lord King."