Is the hadith about the mahdi and dajjal authentic? What will come after the other?
The Mahdi is likely a fabricated addition to Islam. This is the opinion of the respected Egyptian scholar Yusuf al-Qaradawi (as I discuss here).
As for the Dajjal, there are many authentic narrations that mention it, and many narrations of questionable authenticity that add details about it. It seems well-established that there will come a being named Dajjal at the end times due to the number of narrations from multiple Companions that mention it. But the details about the miracles he performs are of doubtful authenticity. Since it is not mentioned in the Quran, I consider it not worth worrying about. But since it is mentioned in many authentic narrations, we cannot reject it entirely until a respected scholar conducts a thorough study and comes up with the conclusion that it is partly or mostly false. For now, we just have to accept things as they are. If the Prophet 
really said these things then we have no choice but to accept them. There also hadith narrations like the following that take away from Dajjal’s importance:
No one asked Allah's Messenger (ﷺ) more about Dajjil than I asked him. He said: He should not be a source of worry to you for he would not be able to do any harm to you. I said: Allah's Messenger, it is alleged that he would have along with him (abundance of) food and water. Thereupon he said: He would be more insignificant in the eye of Allah than that. (Sahih Muslim 2939 a)
The epic narratives that people mention about the end of the world are not mentioned in the Quran, which to me means they are unimportant and not worth concerning ourselves with.