1. The Prophet told us to follow the Quran and Ahlul Bayt:
Hadith Thaqalayn is the simplest and most unambiguous proof of Tashayyu. Hadith Thaqalayn clearly states what our sources of guidance ought to be after the Prophet (s). The Prophet mentions that if we follow the Quran and Ahlul Bayt "we will never go astray."
From Jabir b. ‘Abd Allah who said: I saw the Messenger of Allah, peace be upon him, during his Hajj on the Day of ‘Arafat while he was on his camel, al-Qaswa, delivering a sermon, and I heard him saying: “O mankind! I have left behind over you that which if you hold fast to it you will never go astray the Book of Allah and my offspring, my Ahl al-Bayt.” (Gradings and sources. See also)
2. The Ahlul Bayt are infallible:
Since the Prophet (s) told us to follow the Ahlul Bayt for guidance which we would never go astray with, then they must be infallible, or else this means the Prophet (s) told us to follow a source of misguidance. The Quran mentions in the verse 33:33:
...Allah only wishes to remove uncleanness from you, O members of the (Prophet’s) household (Ahl al Bayt), and to purify you completely. (33:33)
This segment of the verse was revealed upon Muhammad (s), Ali (a), Fatima (a), Hassan (a) and Hussain (a) as mentioned by Hadith Kisa:
Narrated Umm Salamah: "The Prophet (ﷺ) put a garment over Al-Hasan, Al-Hussain, 'Ali and Fatimah, then he said: 'O Allah, these are the people of my house and the close ones, so remove the Rijs from them and purify them thoroughly. (33:33)" (Tirmidhi)
As further proof that that the term Ahl al Bayt only applies to these 5, the Prophet mentioned in the incident of Mubahala:
Narrated 'Amir bin Sa'd bin Abi Waqqas: from his father who said: "When this Ayah was revealed: 'Come, let us call our sons and your sons, our women and your women... (3:61)' the Messenger of Allah (ﷺ) called 'Ali, Fatimah, Hasan and Husain and said: 'O Allah! This is my family. (Ahl)'" (Tirmidhi)
Now, as for the grammar and analysis of the verse and why it necessarily conveys infallibility on the Ahlul Bayt, see the following three articles which explain the verse and objections to the Shia interpretation. (  )
3. The Prophet told us to follow Imam Ali after him and the religion was completed by announcing his successorship:
The Prophet (s) instructed us to follow Imam Ali (a) after his death in a number of places unambiguously. The most obvious mention of Imam Ali's (a) successorship is in the Hadith of Ghadeer which immediately followed Hadith al Thaqalayn where the Prophet said:
It is like I have been called and I will respond. I have left Two Weight Things behind among you. One of them is greater than the other: the Book of Allah and my progeny, my Ahl al-Bayt. Watch how you treat them after me, for the two shall never separate until they meet me at the Pond. (Then he said) Allah is my Master, and I am the master (mawla) of all believers. (Then he held Ali's hand and said) Whosoever I am his master (mawla), this (Ali) is his master (mawla). O Allah, love whoever loves him and be hostile to whoever is hostile to him. (Grading and sources.) (See this video explaining that the meaning of mawla is unambiguously that Imam Ali is meant to succeed.)
Hadith Ghadeer and Hadith Thaqalayn were mentioned by the Prophet (s) after his farewell pilgrimage. Allah had commanded the Prophet (s) to mention these two hadiths in Ayah Tabligh:
“Oh Apostle! Convey what has been revealed to you from Your Lord; and if you do not do so, then you have not conveyed His Message. And Allāh will protect you from the people. Indeed, Allāh does not guide the disbelieving folk.” (5:67)
After the Prophet (s) had mentioned the command to follow the Quran and Ahlul Bayt and the command to follow Imam Ali (a) after him, Allah had revealed Ayah Ikmal al Din:
This day those who disbelieve have despaired of your religion. So fear them not, but fear Me! This day I have perfected for you your religion, and completed My Blessing upon you, and have approved for you as religion, Submission (Islam). (5:3)
(See this discussion on Ayah Tabligh and the subsequent revelation of Ayah Ikmal al Din.)
In other words, the Quran commanded the Prophet to speak the Hadith of Thaqalayn and the Hadith of Ghadeer per Ayah Tabligh, and Allah announced that the religion was completed by these two commands in Ayah Ikmal al Din.
4. Imam Ali is definitively the best of the Sahaba:
Imam Ali (a) is the only person worthy of succeeding the Prophet (s) because he is the second greatest of creation. This is proven by the Hadith al Tayr which says:
From Anas b. Malik who said: Birds were given as gifts to the Messenger of Allah. So, he distributed them and left a bird. Then he said, “O Allah, bring to me the most beloved to You of Your creation to eat with me from this bird. So, ‘Ali b. Abi Talib came and entered and ate with him from that bird. (Gradings and sources) (80 more chains)
There are a few reasons why this hadith is powerful. First of all, it is mutawatir with over 80 chains supporting it. This means it is absolutely established and cannot be questioned. The implication of this is that any hadiths which come with one or two chains which say that Abu Bakr and Umar are better for example are automatically falsified because they contradict this hadith. Therefore, this hadith both proves the superiority of Imam Ali (a) and proves that those who claimed the Prophet (s) said Abu Bakr and Umar were the greatest were liars and fabricators.
Because this hadith is unquestionable by chain standards, some cast doubt on this hadith by saying that the hadith means that Imam Ali (a) is greater than the Prophet (s) and therefore is false. This interpretation is faulty:
1. It is entirely possible to interpret the Hadith as the Prophet calling the most beloved creation after himself. It is far fetched to assume that the Prophet would be asking Allah to call the Prophet to himself
2. Seeing as we have 2 possible interpretations, we can assume for the sake of argument that both are equally strong prima-facie readings of the Hadith
3. The Hadith is mutawatir, and it is therefore almost certain that it was said.
4. So in order to pick the reading that rejects the Hadith, the evidence for that reading must be equally as strong as or stronger than the weight of the Hadith's tawatur (i.e the reading must be certain)
4. But the reading that rejects the Hadith is not certain, and it is not far fetched to assume that the Prophet meant the best after himself
5. So the reading that rejects the Hadith must be rejected because of the prevailing weight of tawatur.
In other words, there is no reason to question Hadith al Tayr, and anything that contradicts it either must be reconciled with it or must be rejected, but it itself cannot be rejected without evidence that is stronger than the weight of its tawatur.
5. There will be 12 caliphs after the Prophet:
The Prophet (s) instructed us that there would be Twelve Caliphs who would succeed him. The Prophet said:
It has been narrated on the authority of Jabir b. Samura who said: I heard the Messenger of Allah (ﷺ) say: Islam will continue to be triumphant until there have been twelve Caliphs. Then the Prophet (ﷺ) said something which I could not understand. I asked my father: What did he say? He said: He has said that all of them (twelve Caliphs) will be from the Quraish. (Muslim)
The reason this proves Tashayyu is because the Prophet (s) describes these caliphs with descriptions that can only apply to us. If for example, I had mentioned to you that tomorrow you will see someone with certain characteristics, that they will wear a red shirt, that they will wear glasses, that they will have shorts on and will wear a blue hat, and a person matching those characteristics shows up- we would naturally say that this is the person that I foretold. The same applies for the description of the caliphs. If the Prophet describes the caliphs with certain characteristics and they only match with Shia beliefs, then it is clear that the Prophet was speaking of Shia beliefs. (See these videos   explaining the argument further).
Shia believe that there will be 12 Imams.
The hadith of 12 caliphs matches this number.
Shia believe that Imam Mahdi will be the last of the 12 Imams, after whom the end of times will come
The Prophet (s) said Imam Mahdi is from the progeny of Fatima (a). (Abu Dawud). He will bring prosperity and justice at the end of times. (Abu Dawud, Muslim). The Prophet (s) said that after the 12th caliph, turmoil will prevail. (Abu Dawud). All of these descriptions match the Shia version of Imam Mahdi (a).
Shia believe that allegiance to the Imam is mandatory and one dies a death of jahiliyyah (kufr) if they do not follow the Imam.
The Prophet (s) said that the one who is not bound by the bay'ah of the caliph dies a death of Jahiliyyah. (Muslim)
Shia believe that obedience to the Imam is absolute and unconditional.
The Prophet said that we ought to obey the caliph and follow their sunnah. (Tirmidhi) Verse 4:59 of the Quran says: obey Allah and obey the Messenger and those in authority among you (4:59). This verse is unconditional, meaning that obedience to them is equivalent to obedience to the Prophet and Allah. (See here). Therefore, they cannot deviate from the commands of the Prophet. As such, they must be infallible. We have already proven that this must be Ahlul Bayt above.
In other words, because all of these descriptions match Shia beliefs, and no other sect can explain all of these hadith as cohesively and simply as the Shia sect can, the Hadith of Twelve Caliphs is best applied to Tashayyu and proves Shiaism.
6. Our own narrative correctly makes prophecies:
The prophecy of two ghaybas
Shias believe that the 12th Imam, Imam Mahdi (a) went into occultation (ghayba). Specifically, we believe he went into two ghaybas, a minor short one, and a long one which we are currently in. Our hadith which were written before the second ghayba mention that the Imam (a) will go into two ghaybas. This prophecy came true. (See this video).
The prophecy of apostacy
The Quran mentions many stories about Bani Isra'il and their failures for a reason. Allah is warning the believers about the fate of our predecessors who even after seeing the miracles of Moosa (a) splitting the ocean and enduring with him on difficult journeys and through plagues and oppression, they still ended up apostating when Samiri built the golden calf. Similarly, the Prophet's (s) companions also saw his miracles, and endured with him in difficult battles and journeys and oppression from the Kuffar. However, the Prophet (s) mentioned:
Narrated Abu Sa`id: The Prophet (ﷺ) said, "You will follow the wrong ways, of your predecessors so completely and literally that if they should go into the hole of a mastigure, you too will go there." We said, "O Allah's Messenger (ﷺ)! Do you mean the Jews and the Christians?" He replied, "Whom else?" (Meaning, of course, the Jews and the Christians.) (Bukhari)
We are the only sect whose narrative fits with the warnings of the Quran and confirms this prophecy of the Prophet (s). On the other hand, the Sunni narrative would not explain this warning by the Prophet (s).