NOTE. parts of my answer is from Shia perspective.
You can find some useful info from Wikipedia on the English, Arabic and Persian entries on the Hadeeth of the two weighty (or precious) things. These pages each has a different content, e.g. the English page shows some Sunni and Shia versions of the Hadeeth, one Sunni version of which is the one you have cited in your question; the Arabic page describes the perspective of the Sunni brothers about the Hadeeth, that it is considered as authentic, but it also adds that the Sunni brothers recognize not only the 14 infallibles of Shi’ite as the household of the holy Prophet --peace be upon him-- but also Abbas and Ja'far and Aqeel and their off-springs as well. The Persian page but beside stating the perspective of Shi’ite about this Hadeeth also cite to a good number of Sunni books that contain the Hadeeth, so I refer you to them listed below for further studying:
- صحیح ترمذی، ج ۵، ص ۶۶۳-۶۶۲، ۳۲۸ عن اکثر من ۳۰ نفر من الاصحاب
- مستدرک حاکم، فصل «فضیلت اصحاب»، ج ۳، ص ۱۰۹، ۱۱۰، ۱۴۸، ۵۳۳،
- سنن ابن ماجه، ج ۲، ص ۴۳۲
- مسند احمد بن حنبل، ج ۳، ص ۱۴، ۱۷، ۲۶، ۵۹، ج ۴، ص ۳۶۶، ۳۷۲-۳۷۰
- فضایل صحابه، احمد بن حنبل، ج ۲، ص ۵۸۵، حدیث ۹۹۰
- خصایص نسایی، ص ۲۱، ۳۰
- صواعق المحرقه، ابن حجر هیثمی، فصل ۱۱، بخش ۱، ص ۲۳۰
- کبیر طبرانی، ج ۳، ص ۶۳-۶۲، ۱۳۷
- کنزالعمال، متقی هندی، فصل اعتصام به حبل ا…ه ج ۱، ص ۴۴
- تفسیر ابن کثیر، ج ۴، ص ۱۱۳، زیر تفسیر آیه ۴۲:۲۳
- طبقات الکبری، ابن سعد، ج ۲، ص ۱۹۴، چاپ لبنان
- الجمیع الصغیر، سیوطی، ج ۱، ص ۳۵۳ و نیز در جلد ۲
- مجمع الزوائد، هیثمی، ج ۹، ص ۱۶۳
- فاتح الکبیر، بنهانی، ج ۱، ص ۴۵۱.
- جامع الاصول، ابن اثیر، ج ۱، ص ۱۸۷
- تاریخ ابن عساکر، ج ۵، ص ۴۳۶
- درالمنثور، حافظ سیوطی، ج ۲، ص ۶۰
- ینابیع المودة، قندوزی حنفی، ص ۳۸، ۱۸۳
But a point that is very interesting to me in this regard is this quotation from the holy Prophet --peace be upon him-- about his household that he says:
From the Arabic page on Wikipedia:
روي عن أبو النصر عن إبن طلحة عن الأعمش عن عطية العوفي عن أبي سعيد
الخذري في حديث عن محمد أنه قال: إنّي أُوشكُ أن أُدعى فأُجيب، وإني
تاركٌ فيكم الثَّقَلَين، كتابَ اللهِ عَزَّ وجَلَّ، وعِتْرَتي، كتاب الله
حَبلٌ ممدود من السماء إلى الأرض، وعترتي أَهْلُ بيتي، وإن اللطيف الخبير
أَخبرني أَنهما لَن يفترقا حتى يَرِدا عليّ الحوض، فَانْظُرُوني بِمَ
تَخلُفُونِي فيهما.
and similar to that from the English page on Wikipedia:
Zaid bin Arqam, narrated that the messenger of Allah . said: "Indeed,
I am leaving among you, that which if you hold fast to them, you shall
not be misguided after me. One of then is greater than the other:
(First is) The book of Allah is a rope extended from the sky to the
earth, and (the second is) my family, the people of my house (ahlul
bait), and they shall not split until they meet me at the hawd, so
look at how you deal with them after me."
And:
From the Arabic page on Wikipedia:
روي عن زيد بن أرقم في حديث عن محمد أنه قال:
قام رسول الإسلام (صلَّى الله عليه وآله) يوماً فينا خطيباً بماء يدعى
خُماً بين مكة والمدينة، فحمد الله وأثنى عليه، ووعظ وذكَّر، ثم قال :"
أما بعد، ألا أيها الناس، فإنّما أنا بشر يوشك أن يأتي رسول ربي فأجيب،
وأنا تارك فيكم ثقلين، أولهما كتاب الله فيه الهدى والنور، فخذوا بكتاب
الله واستمسكوا به " فحَثَّ على كتاب الله ورغَّبَ فيه، ثم قال :" وأهل
بيتي، أذكِّرَكُمُ الله في أهل بيتي، أذكِّرَكُمُ الله في أهل بيتي،
أذكِّرَكُمُ الله في أهل بيتي "
and similar to that from the English page on Wikipedia:
Muhammad said: "One of them (i.e. the Thaqalayn) is the Book of Allah
and the other one is my select progeny (Itrat), that is family
(Ahlul-Bayt). Beware of how you behave (with) them when I am gone from
amongst you, for Allah, the Merciful, has informed me that these two
(i.e., Quran and Ahlul-Bayt) shall never separate from each other
until they reach me in Heaven at the Pool (of al-Kawthar). I remind
you, in the name of Allah, about my Ahlul-Bayt. I remind you, in the
name of Allah, about my Ahlul-Bayt. Once more! I remind you, in the
name of Allah, about my Ahlul-Bayt." [A’alam al-Wara, pp 132-133]
These two quotations very much remind me of the verses [55:7,8,9] of the holy Quran:
وَالسَّمَاءَ رَفَعَهَا وَوَضَعَ الْمِيزَانَ / أَلَّا تَطْغَوْا فِي
الْمِيزَانِ / وَأَقِيمُوا الْوَزْنَ بِالْقِسْطِ وَلَا تُخْسِرُوا
الْمِيزَانَ
And the Firmament has He raised high, and He has set up the Balance
(of Justice), / In order that ye may not transgress (due) balance. /
So establish weight with justice and fall not short in the balance.
I was very much confused about the relations of these verses to each other for a long time, I had heard Ahadeeth about the importance of Surah Ar-Rahman but I just couldn't resolve much about what it says. But then one day I went through its inside meanings by means of interpretation books and I found how these three verses are talking about the Fitnah (in Shia perspective) occurred after the holy prophet --peace be upon him-- passed away. The Firmament in this verse points to the holy prophet and it being raised is addressing his death, the Balance in the verse then points to Imam Ali --peace be upon him-- who is also known as Qasim (divider, that is, the divider of hell and heaven, so that the holy prophet is Abul-Qaseem, Imam Ali's father). I just noted how Allah states that after the holy prophet Imam Ali was set up in the crowd of Muslims for Allah to examine how the Muslims will deal with him after the prophet has passed away. And according to Shia perspective the majority of the Muslims failed to do well, so that you see why the holy prophet before the main content of the Hadeeth first states "إنّي أُوشكُ أن أُدعى فأُجيب" which means "I doubt about if I call you I will be obeyed in response".
EDIT. If one accepts this Hadeeth --which seems to be inevitable-- then the following conclusions seem obvious from it:
Quran is free from any error, be it conceptual or typo, and will be guarded against deviation upto the Resurrection, so would be Ahlul-Bayt --peace be upon them-- infallible, neither ever committing a sin nor ever making a single mistake, and they will never stray. That is, they have been, are and will be "Imams guiding by Our command" (ائمة یهدون بأمرنا) and will never turn to be "Imams inviting to Fire" (ائمة یدعون الی النار). Therefore they are always ultimately trustworthy, and disobeying them would be equal to disobeying Allah and His apostle. And such statements have been independently several timed declare by the holy prophet --peace be upon him and his household-- in several different situations.
There must always be at least one person from the household of the holy prophet --peace be upon him-- who will guide people according to Quran, as Quran will never be left alone among people according to this hadeeth. According to another Hadeeth of the holy prophet --peace be upon him-- whoever doesn't recognize Imam of his time and dies his death would be a death on ignorance. Imam of our time is Imam Mahdi --peace be upon him-- whose name is the name of the holy prophet and whose Kunyeh is the Kunyeh of the holy prophet --peace be upon him.
EDIT.2
The following quotations from the holy prophet --peace be upon him-- is derived from the Qadeer sermon --you can find them in the question asked here-- but there should be other Ahadeeth as well:
30- مَعاشِرَالنّاسِ، تَدَبَّرُوا الْقُرْآنَ وَ افْهَمُوا آياتِهِ
وَانْظُرُوا إِلى مُحْكَماتِهِ وَلاتَتَّبِعوا مُتَشابِهَهُ، فَوَاللَّهِ
لَنْ يُبَيِّنَ لَكُمْ زواجِرَهُ وَلَنْ يُوضِحَ لَكُمْ تَفْسيرَهُ
إِلاَّ الَّذى أَنَا آخِذٌ بِيَدِهِ وَمُصْعِدُهُ إِلىَّ وَشائلٌ
بِعَضُدِهِ (وَ رافِعُهُ بِيَدَىَّ) وَ مُعْلِمُكُمْ: أَنَّ مَنْ كُنْتُ
مَوْلاهُ فَهذا عَلِىٌ مَوْلاهُ، وَ هُوَ عَلِىُّ بْنُ أَبى طالِبٍ أَخى
وَ وَصِيّى، وَ مُوالاتُهُ مِنَ اللَّهِ عَزَّوَجَلَّ أَنْزَلَها
عَلَىَّ.
“O people! Do deeply reflect on the Holy Qur'an, and try to catch the
Verses’ sense and comprehend, then try to observe the even Verses of
its, but do not dazzle at resembled Verses. By God! Whatever is deeply
rooted in it, and the true meaning and the sense of its cannot be well
explained to man, except by this man that I am holding his hand high;
now I announce that: ‘Whomsoever I am master to, so is this Ali to him
master!’ And he is none but Ali, the son of Talib, my brother, my
successor and my help, whose Leadership’s Decree, has been sent to me
from God, Almighty
31- مَعاشِرَالنّاسِ، إِنَّ عَلِيّاً وَالطَّيِّبينَ مِنْ وُلْدى (مِنْ
صُلْبِهِ) هُمُ الثِّقْلُ الْأَصْغَرُ، وَالْقُرْآنُ الثِّقْلُ
الْأَكْبَرُ، فَكُلُّ واحِدٍ مِنْهُما مُنْبِئٌ عَنْ صاحِبِهِ وَ
مُوافِقٌ لَهُ، لَنْ يَفْتَرِقا حَتّى يَرِدا عَلَىَالْحَوْضَ. أَلا
إِنَّهُمْ أُمَناءُ اللَّهِ فى خَلْقِهِ وَ حُكّامُهُ فى أَرْضِهِ.
O people! It is obvious that Ali and my pure descendants born in this
family represent the minor benefit, and the Qur'an represents major.
Either of these two heralds, and relates to another, or they are in
harmony with one another: they will not separate from each other until
they come to me at ‘Koathar’, (on that Certain Day)! Let it be known
that: they are the Trustees of God among His creatures, and are rulers
by His Command on the earth.
61- مَعاشِرَالنّاسِ، إِنَّ اللَّهَ قَدْ أَمَرَنى وَنَهانى، وَقَدْ
أَمَرْتُ عَلِيّاً وَنَهَيْتُهُ (بِأَمْرِهِ). فَعِلْمُ الْأَمْرِ
وَالنَّهُىِ لَدَيْهِ، فَاسْمَعُوا لاَِمْرِهِ تَسْلَمُوا وَأَطيعُوهُ
تَهْتَدُوا وَانْتَهُوا لِنَهْيِهِ تَرشُدُوا، (وَصيرُوا إِلى مُرادِهِ)
وَلا تَتَفَرَّقْ بِكُمُ السُّبُلُ عَنْ سَبيلِهِ.
“O people! God has stated me His exemption and restriction of certain
things, and I also, by Divine Command, have ordained Ali the knowledge
of it, and forbidden him [some things]. And [thus] the knowledge of
ordaining and forbidding, remains with him; therefore, listen to his
commands, so that to be safe, and obey him so as you may get on the
course. Accept his Warning, so that to find the right direction, and
walk on towards his ultimate destination! [Thus, beware] so as not to
draw away from His Path into the diverted paths!” (Livestock: 6/153).
Godspeed.
