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In Japanese, you have both politeness levels and formal levels, it can be easy to mix them up. じゃ is fine in politeness, but lacks in formality. Since you don't have to be formal with your teachers, じゃ is fine (and yes, teinei form is used towards teachers unless it's like... a professor or something where sonkeigo can be fitting).

The politeness levels differs a whole lot, from grammar to completely new words. For example, 見る、見ます、拝見いたします、ご覧になる. Those all mean the same thing, but are from different politeness levels.
見る = non-polite, casual
見ます = teinei
拝見いたします = kenjougo, used by the person of lower status in a conversation about himself
ご覧になる = sonkeigo, used by the person of lower status in a conversation about the person of higher status

I'm not sure how helpful this will be, but it's been personally useful in understanding what Tobberoth was talking about in terms of English. I'm just gonna expound on what he said.

You can think of keigo in terms of three levels or ways of talking, these are honorific expressions, modest expressions, and humble expressions.

Honorific expressions exalt the subject of the sentence, and are never used to refer to what one does (I do). This includes expressions like "goran ni naru", "meshiagaru", "oyasumi ni naru". This is talking about other people highly.

Modest expressions are the exact opposite; they play down whatever the speaker does and, obviously, sound modest. That's like "haiken itashimasu", "mairimasu", "itadakimasu", etc. This is talking about yourself modestly/lowly.

finally, humble expressions are sort of like both of these; they lower the speaker's position and raise another persons (not necessarily the listener). These are o+verb stem+suru verbs, for example,
"Watakushi ha sensei ni hon wo okashi shimashita/itashimashita. " "I (humbly) lent the book to the professor".

I hope this helps/ makes sense :P

Thanks! ^^

Posted By: Jumex

[p]I'm not sure how helpful this will be, but it's been personally useful in understanding what Tobberoth was talking about in terms of English. I'm just gonna expound on what he said.[/p][p]You can think of keigo in terms of three levels or ways of talking, these are honorific expressions, modest expressions, and humble expressions.[/p][p]Honorific expressions exalt the subject of the sentence, and are never used to refer to what one does (I do). This includes expressions like "goran ni naru", "meshiagaru", "oyasumi ni naru". This is talking about other people highly.[/p][p]Modest expressions are the exact opposite; they play down whatever the speaker does and, obviously, sound modest. That's like "haiken itashimasu", "mairimasu", "itadakimasu", etc. This is talking about yourself modestly/lowly.[/p][p]finally, humble expressions are sort of like both of these; they lower the speaker's position and raise another persons (not necessarily the listener). These are o+verb stem+suru verbs, for example,

"Watakushi ha sensei ni hon wo okashi shimashita/itashimashita. " "I (humbly) lent the book to the professor".[/p][p]I hope this helps/ makes sense :P[/p]