Salamun 'alaykum,
my background is Sunni and I have to admit that there were a lot of things that I would hear or see from people, who regard themselves as Shi'a, that would make me not even consider Tashayyu' (Shi'ism) to be true and this despite always having huge respect and love for Amir al-Muminin 'Ali bin Abi Talib (peace be upon him).
What made me change my mind is listening to the Shaykh Hasan bin Farhan al-Maliki and reading his articles.
That's why I would like to give my Shi'a brothers, who are upon the Madhhab of al-Imamiyya / al-Ithna 'Ashariyya (considering that 12ers constitute the majority of Shi'a today) some advice in order not to end up being a reason for people to stay away from Tashayyu':
1. Commemorating the Istishhad (martyrdom) of Imam al-Hussayn (peace be upon him) and that of others from the Ahl al-Bayt (peace be upon them) in a manner that people understand the reason behind these tragic incidents and the lessons for us to learn from them.
A lot of Shi'a today unfortunately think of it as some sort of ritual, where you have to beat your self in a specific beat or crawl on the ground or do other weird actions and that's it. The outsider will only see these weird actions and not really understand what the matter actually is (like standing up against oppression).
2. Highlighting the importance of depending on Allah ta'ala in all situations and that Tawassul through the Prophet (sallallahu 'alayhi wa alihi wa sallam) and his pure Progeny is legistaled and not going overboard.
While there is no doubt that asking Allah ta'ala through the Wasila of our noble Prophet (sallallahu 'alayhi wa alihi wa sallam) and his pure Progeny is legislated and has been acted upon by pretty much everyone prior to the emergence of the Najdi extremists (who relied upon the claims of the anthropomorphist Nasibi Ibn Taymiyya), this does not mean that one should call upon them in a manner that is only befitting for Allah ta'ala nor to make ones heart depend on Asbab and become heedless of our Creator jalla jalaluhu.
3. Highlighting the importance of staying chaste and staying away from sexual immortality and not forgetting the wisdom behind rulings.
Some Shi'a today will defend temporary marriage as if it's the most important thing in the religion, not knowing that even fellow Shi'a (like the Zaydiyya) may disallow it and forgetting that rulings have a wisdom behind them and that even a permissible thing may lead to sinfulness if taken out of its correct context of application and misused. Take polygamy as an example: If a person marries more than one woman and ends up being injust to one [or all] of them, then he will be sinful despite the contract itself being correct. The Shaykh Ahmad al-Waeli mentioned that Mut'a should only be done, if there is a necessity and he gave an example: A person travels to a land to study for a long time and enters into such type of marriage to protect himself from immorality.
I mentioned the above three points, because these three directly come to the Sunni mind and are quite off-putting in that they lead to not even consider Tashayyu' as the correct view.