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.
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.