What verses in the Quran tell us to follow the Hadith (of the Prophet) or guide us to do so?
There are many verses that say a believer should obey (أَطِيعُوا) the prophet: 3:32, 3:132, 4:59, 5:92, 8:1, 8:20, 8:46, 24:54, 47:33, 64:12. So if the prophet orders a believer to do something that person should do it (otherwise it is disobeying God).
However the issue of following hadith is complicated for various reasons. Three main ones are:
It is not easy to determine if a hadith is authentic and accurately narrated. Sometimes people made up hadith. Sometimes people didn't accurately narrate a hadith.
Even in the case of authentic accurately narrated hadith the intention is not always clear. Was it an order to a specific person that does not generalize or was it a general order to all believers? If it generalizes how does it generalize to different situations? Was it an order or an advice?
Even in the case that we determine it was an order that generalizes, we may need further information about the details of carrying it out.
Therefore it is not an straightforward to say you should follow hadith or you don't need to follow hadith. Studying these issues in a systematic holistic manner (in the context of other information, e.g. Quran itself) to find answers to these questions is one of the main things Muslim scholars do.
What is certain is that if you are certain about all of the following you should follow it: it is authentic and accurately narrated, it gives an order that applies to your situation, and the details of how to carry out the order are clear.
The other answers seems to be answering on the definition of the word hadith, which seems not to be the fundamental question. As many words, the meaning of the word differ according to the context. The verses stated by Sayyid about the word Hadith in the Quran has no connection to your question.
This could be answered in different ways according to once understanding of the question. If you are looking for a statement in the Quran that directly, and with the literal words says for instance, "Follow the hadiths!". Then, the answer would be no, there are no such verses.
But this still doesn't mean that we shouldn't follow the sunnah like some might argue. You cannot follow the sunnah without the hadiths, because that's where you get the data, after you gotten the data, you can fetch it and analyse it. That is what we later will define as "sunnah" or the way of the Prophet.
What does it even mean to follow the sunnah? Are there any legitimacy in claiming that one should follow the sunnah?
In my opinion (and the vast majority), yes there are good claims that one should. If one analyses Islam deeply, you would find it a logical fallacy to ignore the hadiths or the sunnah of the prophet. Now, I do not in any way state that you aren't following the sunnah or the hadiths. But as the question tends to lean towards that understanding I just feel it should be answered more in detail.
In the verse 4:59 we find:
O you who believe! obey Allah and obey the Messenger and those in authority from among you; then if you quarrel about anything, refer it to Allah and the Messenger, if you believe in Allah and the last day; this is better and very good in the end.
The small group of people that today explain verses like these argue that this really mean to follow the message and not really the messenger. But that statement is fast thrown away if you read through "and those in authority from among you". It's clear that it's talking about a person with authority.
The prophet is the one that would explain for those who would "quarrel about anything", as you see in the verse. Today we cannot directly ask the prophet to explain what certain verses meant or how to act in certain situations. But if we look at the hadiths, with the Quran we can draw different conclusions according to the situation, time and context.
I do not think anybody would say that they wouldn't follow the Prophet if he was here and could explain things to them, or if he ordered/forbidden something that the Muslims would be against that, I don't think that. But I do understand that many feel that, because of he isn't here and really cannot explain things, some would throw away the sunnah (which are derived from hadiths). I also think this is done of fear and ignorance.
Just because a person doesn't understand a hadith doesn't mean it is wrong. We know that the prophets are sent with the message, and they are in at the same time the leader, mentor, and guide of their people. We should follow the prophet. How to understand what to follow is something else and another discussion. There are many hadiths out there that make no sense without the tools of understanding. Sometimes to truly understand a hadith you must need to understand Arabic (deeply), that includes figure speeches, expressions and much more. The context, to whom was this hadith said? Why? When? Where? It's something to leave to the experts.
And as many people might not know, there are some hadiths that are classified as
Mutawatir. The Quran is classified as
Mutawatir. To throw away all Mutawatir hadiths would be like literally know that the prophet said something and really say that he didn't say that. If a person argues it's not trustworthy then how does he think the Quran is thustworthy? I mean there should be some kind of limit.
When a person knows about the mutawatir hadiths and the way of collecting them, it will be very hard to deny them.
Say: If you love Allah, then follow me, Allah will love you and forgive you your faults, and Allah is Forgiving, MercifuL
If you think how it developed by the time it will look more logically. For instance, the companions follow the Prophet (saw). When he passes away, they spread his message to whoever they can, they speak about him, what he have said, what he have forbidden and so on. And this process keeps going, generation to generation, thankfully, or we might not been Muslims now I guess. To ignore this tradition is more likely to be wrong. I mean, 1400 years later we finally understand that we should not follow the hadiths or the understanding of them by our scholars.
The answer might be a bit big but I really think it is necessary in this case. Much more could be said, I guess a book could be written about the subject.
Hadith just means a narrative, a story, a speech, a communication...Many verses tell us to follow Hadith, but they are all referring to the Quran since the Quran is called the "Best/aHsan Hadith" (Quran 39:23).
You can read some other verses:
"Allah - there is no deity except Him. He will surely assemble you for the Day of Resurrection, about which there is no doubt. And who is more truthful than Allah in HADITH?" (Quran 4:87)
"Do they see nothing in the government of the heavens and the earth and all that Allah has created? (Do they not see) that it may well be that their terms are drawing to an end? In what HADITH after this will they then believe?" (Quran 7:185)
"These are verses of Allah that We recite to you with truth. Then, in what HADITH after Allah and His verses do they believe?" (Quran 45:6)