A verb ta-form followed by ほうがいい means “had better” or “should do something” used when making suggestions or giving advise. 気をつけた is the ta-form of 気をつける meaning “watch out” or “be careful” so 気をつけたほうがいい means “You’d better be careful”.
The sentence ending particle よ is used when telling something the speaker thinks the listener should know so it’s often used when giving advice.
もう means “already”.
始まってる is the shortened form of 始まっている. A verb te-form followed by いる often used with もう means “have already done”. It’s used when actions have already been done before a specific point in time. So in this case with もう始まってる the action “始まる” has been done before the moment they start talking which is the present. So もう始まってる means “it has already begun”.
かもしれない means “may” or “might”.