Learn Hardcore Turkish: Telefon numaralarını not etmek için kalem ve deftere ihtiyacımız var. - We need a pen and notebook to note down the phone numbers.

-mak/-mek turns a verb into its infinitive (“to do something”). Adding için after that infinitive expresses “in order to” or “for the purpose of.”
In our sentence:
not etmek için = “in order to write down” (literally “to do ‘note’ for the purpose of”).

Why do we say not etmek instead of just using not almak or kaydetmek?

not is a borrowed noun meaning “note.” When you combine it with etmek (“to do/make”), you get the verb not etmek = “to make a note of,” “to note down.”
not almak is also possible (“to take a note”), but not etmek is more common for quickly jotting something down.
kaydetmek usually means “to record,” often in a broader or digital sense, not just a quick manual note.

Why is telefon numaralarını in the accusative case (–yı suffix)?

Turkish marks definite direct objects with the accusative suffix –ı/–i/–u/–ü (with a buffer consonant y or n when needed). Here, telefon numaraları refers to a specific set of numbers, so we mark it as a definite object:
telefon numaraları + yı → telefon numaralarını
(The vowel harmony and buffer consonant turn –yı into –nı after the final –ı.)

Why is there only one dative ending (–e) on deftere, but none on kalem?

When you have two (or more) nouns joined by ve (“and”) that share the same case, Turkish lets you attach the case ending only to the last noun.
So:
kalem ve deftere
both are in the dative (“to a pen and to a notebook”), even though only defter carries –e.

How does ihtiyacımız var translate as “we need”?

ihtiyaç = “need” (noun)
Attach the possessive –ımız = “our” → ihtiyacımız = “our need”
Add the existential verb var = “there is/are”
Literally: “our need exists”, which we interpret as “we need.”

Where is the English “a/an” before kalem and defter?

Turkish does not have an indefinite article like a/an. You can optionally use bir for emphasis (“one”), but omitting it still conveys “a pen and a notebook.”

Why isn’t telefon in the genitive case (like telefonun numaraları)?

telefon numarası is a common compound in Turkish, so you don’t need the linking genitive –(n)ın. Saying telefonun numaraları would be grammatically correct but sounds more awkward or overly literal (“the numbers of the phone”) compared to the fixed phrase telefon numarası.