Resolved questions
How many different ways can you say "I" in Hebrew?

אני should get you covered. It is used wherever you'd say "I" bar when it's in the object.

Do notice:
1. It might be omitted in simple sentences, mostly when it precedes a verb in the past tense (which imply the subject). I recommend you use it anyway - better safe than sorry.
2. אני can also replace "me" when it is (a part of) the subject. However, "me" (or "I") as the object in hebrew takes the form of an inflected preposition. Caution is advised. :-)

Okay, so there's also אנוכי , used most famously in the first commandment. Basically it's no longer a part of modern hebrew. Today it's used rarely, mostly to set a poetic or spiteful tone.
It also means "selfish", which is the more probable meaning should you encounter this word.

דוגמאות:
שלום, אני ג'סי.
אני חושב משמע אני קיים
מי שמביט בי מאחור לא יודע מי אני (sesame street)
אני אוהב שוקולד ועוגות גבינה (famous children's song)

אנוכי יהוה אלהיך - I am the lord your god
זה כל-כך אנוכי מצדך - That's so selfish on your part (literally "from your side")