Levels of respect in speaking
Are there only three levels of respect?
Honorific, Formal/Polite, and Informal/Casual ?

For example:
1. 미안합니다
2. 미안해요
3. 미안해

1 I would use with elders/strangers/anyone older than me who I don't know.
2 I would use with teachers? Or even adults I know? People from school who I'm not necessarily friends with?
3 Friends and family?

It's hard for me to distinguish where the line is drawn, I know everyone has a different relationship with people in their lives, but as a general rule to begin with? :)

Question

Levels of respect in speaking
Are there only three levels of respect? 
Honorific, Formal/Polite, and Informal/Casual ?

For example:
1. 미안합니다 
2. 미안해요
3. 미안해

1 I would use with elders/strangers/anyone older than me who I don't know. 
2 I would use with teachers? Or even adults I know? People from school who I'm not necessarily friends with?
3 Friends and family?

It's hard for me to distinguish where the line is drawn, I know everyone has a different relationship with people in their lives, but as a general rule to begin with? :)

Answer

There are seven verb paradigms or speech levels in Korean, and each level has its own unique set of verb endings which are used to indicate the level of formality of a situation. 
Hasoseo-che  (하소서체)
Formality	very high
Politeness	high
archaic
ex) 미안하나이다

Hapsyo-che  (하십시오체)
Formality	high
Politeness	high
ex)미안합니다.

Haeyo-che(해요체)
Formality	low
Politeness	high
ex) 미안해요.

Hao-che(하오체)
Formality	high
Politeness	neutral
Currency	 uncommon
ex) 미안하오.

Hage-che(하게체)
Formality	neutral
Politeness	neutral
older generation
ex) 미안하네

Haera-che(해라체)
Formality	high
Politeness	low
ex) 미안하다

Hae-che(해체)
Formality	low
Politeness	low
ex) 미안해

Refer to https://en.wikipedia.org/wiki/Korean_speech_levels

1. 미안합니다 is Hasipsio-che
This conversational style is generally called either the "formal" or the "formal polite." This is a common style of speaking. A conversation with a stranger will generally start out in this style and gradually fade into more and more frequent haeyo-che. It is used
between strangers at the start of a conversation
among male co-workers
by TV announcers
to customers

2. 미안해요 is Haeyo-che
This speech style is called the "polite" style in English. It is remarkable in that it is used both with higher level pronouns. It is used mainly:
In Korean phrasebooks for foreigners.
Between strangers, especially those older or of equal age.
Between female co-workers or friends.
By younger speakers as a less old-fashioned alternative to the hao-che.
By men and women in Seoul as a less formal alternative to the hasipsio-che.

3. 미안해 is Hae-che
This conversational style is called the "intimate" in English. It is a defective paradigm, meaning it lacks most of the expected conjugational forms. It is used
Between close friends and relatives.
When talking to children.

You can use Hasipsio-che for anyone older than you.
You can use Haeyo-che for younger strangers/anyone not intimate same age with you.
You can use Hae-che for children/intimate friends/family

Answer

I'm really interested in this, too! Maybe Tony can help us, he has helped me a bunch twice with my simple Korean questions :)