For some reason I feel that a noun at the beginning of a sentence has to have the article. Obviously though I'm not anything close to an expert, am I totally mistaken about that?
You are partly right!!! Usually the first noun is the subject of the sentence and it has article, but this is not a typical Greek sentence as we don't have the Subject-Verb-Object form. Therefore, the article is not needed here.
Does the word πιπέρι mean both the spice peper and the vegetable peper or are there different words for each in greek?
It's το πιπέρι for the spice and η πιπεριά for the vegetable.
In Cyprus we would often use "άλας" (not sure of the spelling) for salt is this correct or is it just slang for Cyprus
Not sure if it's slang in Cyprus, but it's definitely very formal in Greece.