All cases in Greek (and in English, for that matter) have some sort of sign by which they can be recognized. In English, for example, the sign of the possessive case is "s." That's how we know that "his" means "of him," for example, while the "m" of "him" indicates that "he" is an object, not a subject.
So in Greek, the sign of the dative case is an iota lurking somewhere at the end of the word. In ἀνθρώπῳ, for example, it dangles under the final omega. In Ἕλλησι, it's right at the end of the word.
The dative case, here as (usually) elsewhere, indicates to or for whom (or which) something is said, given, or done. In this case, it's "to the Hellenes."
On to verbs:
The aorist indicative is a simple past tense. Most verbs have a first aorist, some verbs (like παρέχω) have a second aorist. You have to know where, in your Greek grammar, to look for the conjugations of each. How do you know which verbs have a second aorist? Because when you learn a new verb, you learn its principal parts. You can conjugate the English verb "go" in all tenses because you know its principal parts: go, went, gone. Similarly, when you learn the very common verb ἔχω, you learn its principal parts: ἔχω, ἕξω, ἔσχον. And bingo, you see from that -ον ending that it has a second aorist.
Don't be discouraged. Get some sleep. Go to class and ask questions, no matter how painful it is. Understand that by learning Greek you're learning more about your own language, and understanding the power that comes from clarity of thought.