Learn Russian: Нет, сахара нет, и соли тоже нет. - No, there is no sugar, and there is no salt either.

It serves two different functions! The first нет simply means "No" (the opposite of "Yes"). The second and third нет mean "there is no". In Russian, you use this second type of нет to state that something is missing or absent.

Why are the words for sugar and salt written as сахара and соли?

Because they show what is missing after нет ("there is no"), they must be in the Genitive case. Сахар (sugar) is a masculine noun ending in a consonant, so we add -а to get сахара. Соль (salt) is a feminine noun ending in a soft sign (ь), so the -ь changes to -и to get соли.

I thought тоже meant "also" or "too". Why is it translated as "either" here?

Тоже does mean "also" or "too". Literally, the second half of the sentence translates to "and of salt there is also none." However, in English, when we agree with a negative statement, we use "either" instead of "also." Russian keeps it simple and just uses тоже.

Is it okay to say нет сахара instead of сахара нет?

Yes, both are perfectly fine! Russian word order is very flexible. Putting the items first (сахара нет, и соли тоже нет) puts a little more emphasis on the sugar and salt themselves, as if you are checking off a list: "Sugar? None. Salt? None either."