다다음 주부터 가을방학 시작한다.
The examples in my book don’t use a particle next to school vacation. I also wasn’t sure if “다다음“ is week after next? Is there a different informal way to say “in two weeks”?
It is correct but not the most natural way to say it.
Compare these related sentences.
시작한다 in #1 is typically used as a transitive verb requiring an object. 시작된다 is its intransitive version. The simplest and most common way to say it is with 이다 (is) though, like #3.
Also, phrases like 가을방학 시작한다 or 가을방학 시작된다 sounds abrupt and unnatural because of the missing particle. Particles can be dropped in a lot of phrases but there are also many cases where the omission affects the natural flow of the sentence, like your example of 가을방학 시작한다. When you're not sure, it's best to include the particle.
다다음 주 is "the week after next". "in two weeks" is 2주 후에. You can also say 2주 있으면 (after two weeks, two more weeks and (...). The particle 면 creates a condition a particular outcome depends on). For example, 2주만 있으면 가을방학이다 = Only two more weeks and it is fall vacation, which is probably the most commonly heard phrase of this kind.