始まる is the intransitive verb while 始める is transitive verb.Therefore... その冒険は始める is incorrect, you are to say that その冒険を始める その冒険は始まる might be no problem, however, mostly particle が is used in common phrase. Transitive verb: The sentence can take particle を with the subject in particle は. 　（わたしは 〇）その冒険を始める Intrasitive verb: The sentence can not take particle を, but が instead wihout the subject in particle は. But you can use particle は when it is topic maker. 　（わたしは ✕）その冒険が始まる - In this sentence, わたし is the subject. That's why it doesn't make a sense. 今日は、その冒険が始まる - In this sentence, 今日 is not the subject but topic. That's why it is no problem.

"始める" is to start by your decision or thought. "始まる" is to start independently of your opinion and thought.