2. "Скучать" (without any object! It is intransitive) as "to feel lack of any interest". It can also be related to some melancholy, but not always.

Я скучаю. - I feel bored. I am not interested in anything right now. I do not want to do anything. (Maybe also, but not necessarily: I am sad because of that). Anyway, it's a negative feeling.

Я скучаю на этой работе. - In this case, the reason is specified indirectly. I feel bored at this work. (Probably because this work is uninteresting for me).

Почему ты скучаешь? - Whay are you so bored? (I see your sad face, I see you are not doing anything).

Again, no direct object is possible. And no infinitive is possible either. One cannot say "Я скучаю + что-то делать". It is ungrammatical, and it is even illogical: if he is already doing something, then he is not bored.

There are also 2 related words:
Скучно is an adverb of state.

Мне скучно. = Я скучаю.
Мне скучно здесь без тебя. = Я скучаю здесь без тебя.

But with "скучно", you CAN use an infinitive:
Мне скучно с тобой разговаривать. = I feel bored of talking to you (because it is uninteresting to me, talking to you makes me sleepy).
Здесь скучно работать. = It is boring to work here (because the work is uninteresting, moreover it is monotonous, the same routine every day).
Скучно быть одному. = It is boring to be alone.

Скучный - adjective. = boring, uninteresting, monotonous: скучная книга (boring, uninteresting book); скучная работа (boring, monotonous job), скучный фильм, скучный вечер (you had nothing to do that evening).

Now, let's compare "скучать/скучно" and "надоесть". The former expresses boredom, the latter expresses tiredness:

"Мне скучно с тобой разговаривать" - It is boring, uninteresting for me to talk to you. Certainly, it does imply that I do not want to continue talking. But the reason is the lack of interest.

"Мне надоело с тобой разговаривать" - I am tired of talking to you. It does not necessarily mean that "мне скучно". It can, but not always. It just states: "I do not want to continue talking to you" but there can be different reasones: it may be uninteresting; it may be interesting but I've just got tired (I need some break); maybe I do not like the way you talk; maybe I dislike you personally etc.

"Мне скучно" can be said when you are doing nothing, and do not want to do anything due to your emotional state. It does not mean that something "надоело", you are not tired of anything, you are just bored.

"Мне скучно работать здесь" - You feel bored of working here, because the work is monotonous and uninteresting. It does not directly mean you want to change your job (but it is quite logical, although it is unexpressed)
"Мне надоело работать здесь" - You are tired of working here and do not want to. It definitely states you would like to change your job. But it does not mean "мне скучно". Maybe yes ("надоело" because" of "скучно") but maybe not ("надоело" because the work is hard, difficult, or because the boss is always angry, or because the salary is low etc.).

Consider 2 dialogs:
- Я завтра увольняюсь с работы.
- Почему?
- Потому что эта работа скучная. Она мне надоела. Я больше не могу там работать.
(Option: Потому что мне скучно на этой работе. Мне надоело там работать. Я больше не хочу.)
(Another option: Потому что мне надоела эта работа. Я скучаю на ней. Я хочу другую работу.)

Here, "надоело" because of "скучно".

- Я завтра увольняюсь с работы.
- Почему?
- Она мне надоела.
- Что, работа скучная?
- Да нет, не скучная. Я просто устал. Начальник всё время кричит на меня. Я ничего не успеваю. Куча заданий. Надоело!

Here, "надоело" but not "скучно"!