How do I know when to use infinitive 다 form when speaking very casually to someone?
So in a TTMIK book called Real Life Conversations: Intermediate two close friends said to each other over the phone:

A: 아, 그렇구나. 그래야겠다. 조만간 다 같이 한번 보자.
Oh, I see. I should. Let’s meet up all together one of these days.

B: 그래. 한번 뭉치자. 본 지 진짜 오래 됐다.
Sure. Let’s get together. It’s been a really long time since we last met.

Particularly in the last sentence of what person B said they used the infinitive 다 form of the verb 되다. Does it make any difference in meaning when keeping it in this form? Does it show that you’re saying it to yourself? How do I know when I can use this form in casual conversation? And does saying a causal sentence like this in the infinitive form add any underlying nuance to the sentence?

Question

How do I know when to use infinitive 다 form when speaking very casually to someone?
So in a TTMIK book called Real Life Conversations: Intermediate two close friends said to each other over the phone:

A: 아, 그렇구나. 그래야겠다. 조만간 다 같이 한번 보자.
Oh, I see. I should. Let’s meet up all together one of these days.

B: 그래. 한번 뭉치자. 본 지 진짜 오래 됐다.
Sure. Let’s get together. It’s been a really long time since we last met.

Particularly in the last sentence of what person B said they used the infinitive 다 form of the verb 되다. Does it make any difference in meaning when keeping it in this form? Does it show that you’re saying it to yourself? How do I know when I can use this form in casual conversation? And does saying a causal sentence like this in the infinitive form add any underlying nuance to the sentence?

Answer

The -다 ending is one of the common sentence ending forms.  It is not the infinitive form (infinitive forms are very rarely used).

Compare the ending forms for verbs and adjectives (되다(v) and 좋다(a) as examples). 
* -ㅂ니다: formal and polite (to someone older, in public speech). 됩니다, 좋습니다. 
* -다: impersonal and neutral/casual (in dry written material, between close friends, to children). 된다, 좋다.
* -아/어요: social, polite (default form between adults in everyday settings). 돼요 (from 되어요), 좋아요.
* -아/어: informal, familiar (between close friends). 돼, 좋아.

(For the present tense, the adjective ending(좋다 in this case) becomes the same as the root form.  A verb in the present tense (된다) is not the same as the root form (되다).  This is one of important differences in verb/adjective usage.)

The sentence endings don't change the meaning.  They only change the politeness, formality, and nuance.

The -다 ending can have these nuances relative to the more common -아/어(요) endings.
1. surprise or enthusiasm as in an exclamation.
2. giving an objective fact (drawing attention to it).
3. talking to oneself (there are better forms for this though).

Example:
A: 뭐 찾아?  (-아/어 ending sounds the most normal with personal and social feel)
B: 응, 내 차 키가 안 보여서 (찾고 있어).
A: 잘 찾아 봐. 어디 있겠지.
B: 아! 찾았다! 탁자 옆에 떨어져 있었네. (-다 here is more of an exclamation (to both oneself and the listener) than part of the dialogue.  찾았어 would retain the normal calm tone and won't carry the same kind of surprise.)