이 겨울이 새로운 탄생의

첫 발임을 굳게 믿는다.

이(this) 겨울이(winter) 새로운(new) 탄생의(of a birth)

첫(first) 발(foot, step)임(the fact it is; it being)을 굳게(firmly) 믿는다(believe).

So it means I firmly believe that this winter is the first step for a brand new birth.

첫 발임을 might be a little tricky to understand.  You may think of it as two phrases.

- 이 겨울이 새로운 탄생의 첫 발이다 - This winter is the first step for a new birth.

- (나는 이것을) 굳게 믿는다 - I firmly believe (this fact).

=> (나는) 이 겨울이 새로운 탄생의 첫 발임을 굳게 믿는다.

* 첫 발이다 -> 첫 발임(을):  is the first step -> that it is the first step / being the first step (turns a full clause into a noun phrase).

Here's my two cents

1st Subject: 나는 (I - which is omitted in this case)

2nd Subject: 이 (this) 겨울 (winter)

1st Verb: 믿는다 (believe)

2st Verb: 이 (is)

Object: 새로운(new) 탄생 (birth) 의 (of) 첫 (first) 발(step,foot)임(nominalization) 을 (an object marker) 

Adverb for 1st Verb: 굳게 (firmly) 

이 (this) 겨울 (winter)이 (is)새로운(new) 탄생 (birth) 의 (of)

Translation: I firmly believe that this winter is the first step of a new birth.