The most common uses of -지 are these:
1) saying something with the tone of confirmation or gently repeating what's already known.
2) asking about something partially known for confirmation.
* 나는 한국어를 공부하지 is like "You know what. I am studying Korean".
Compared with the -아/어 ending, -지 sounds a little like saying something to oneself, like boasting about it.
* -아/어야지 is a contraction of -아/어야 하다 connected with the -지 ending. -아/어야 하다 is the same as -아/어야 되다, so the difference comes down to the -지 ending. The meaning varies slightly with the sentence type (statement v question) and the person (1st, or 2nd/3rd).
- 친구하고 사이좋게 놀아야지 = gentle urging, like "You should ...".
- 난 열심히 공부해야지. = expressing one's will, like vowing to oneself.
- 너 내일 학교 가야지[가야 하지/되지]? = confirmation question.
* 그래 (from 그러해 or 그렇게 해) = 1a) Yes, it does/is. 1b) Yes, go ahead (permission). 2) Is that so?
* 그지 (= 그렇지) = 1) Yeah, I bet (informal confirmation), 2) That is so, isn't it? (confirming question)
Meaning-wise 그래 and 그지 are very close, but 그래 is ordinary while 그지 is more informal and implies it's something already known, because -지 imparts the sense of a knowing/agreeing conversation.
The -지 ending cannot indicate giving permission the way 그래/그러세요 can.
- 날씨 좋다. 그지? = It's a great day, isn't it?
* The general principle 이/저/그 designation is: 1) 이 = close to the speaker or both interlocutors, 2) 저 = away from both people, 3) 그 = a) away from the speaker and close to the listener, b) referring back to something already mentioned.
- 이렇게 하고 싶어요 = indicates something one is doing now or about to do.
- 그런 사람을 믿지 말아요 = such type of people. (the type must have already been mentioned)
- 저런 사람들 = "those people" - they are in view at a distance.
Suppose your friend showed up with someone you don't know. You might ask 그분은/이분은 누구셔? (그 if there's some distance to them). But if the two of you saw someone passing by in the distance who looks like another friend, you'd say 야, 저 여자/쟤 [name] 같다.