Here's some old threads that touch on your question and may be of interest. They're mostly about Roman history.

https://www.reddit.com/r/AskHistorians/comments/21pgk2/why_does_it_seem_that_the_roman_persecution_of/?

https://www.reddit.com/r/AskHistorians/comments/3obynd/why_did_the_traditionally_relatively_religiously/?

https://www.reddit.com/r/AskHistorians/comments/2dq8qb/what_is_the_historicity_of_early_christian/?

There's too much going on in your question overall to give it a complete answer. (1) It would be foolish for me to try and answer for whatever your parents may or may not have told you, (2) Dealing with persecution of Christians throughout history is too broad a topic.

Were Christians persecuted and killed for being Christians in the Roman Empire?

Yes. Not always, not everywhere, and not necessarily as people think, but it did happen.

One of our earliest documentary evidences for Christians outside and after the New Testament are the letters between Pliny the Younger, during his time as governor of Bithynia, and Trajan the Emperor. These are letters 10.96-97, written around 112 AD, in which Pliny asks advice for what to do with Christians. The correspondence makes clear that while active hunting of Christians is not advocated, Christianity itself is seen as punishable, not merely Christianity + other offences.

Persecution in the Roman Empire was generally localised, and this reflects the fact that administration was localised. Even when you get edicts seeking to force Christians to sacrifice to the genii of the Imperial family and the Roman gods, the actual enforcement of those edicts appears to vary in severity. Significant persecutions did occur under Decius and Diocletian.

Martyr accounts, when they began to be written in the mid 2nd century, had a strong tendency to model themselves on earlier Christian texts, including the accounts of Jesus' death. This is seen already in the death of Polycarp, which explicitly tells you that it is modelling itself on Jesus' death. However, contra Moss, the fact of literary styling shouldn't by itself rule against the historicity of the text. Nonetheless, as martyrdom increases in its theological significance, as can be seen for instance in the emergent theology of martyrdom in North Africa in the 3rd century in Tertullian, Cyprian, and Perpetua's martyrdom, this leads to the undoubted development of martyrdom accounts as a literary genre ripe for ahistorical 'hype' and fictionalisation.

Since you mentioned China (where I live), I'd like to reprint an answer I gave to u/FlyingRowan a few months ago, who asked "Genuine question: growing up as a Christian I was told every day that Christians were actively hunted and murdered in China... Is there any validity to what I was told?"

This was my reply: Short answer: No, it's not true.

Longer answer: It was a little true in 1900 and 1966-76. Other than that, Christians faced some problems, with some leaders being arrested, but not things like being murdered.

Full answer: In the Qing period, Chinese Christians were often the focus of anti-foreigner sentiment, which could occasionally include riots and murders. The most significant event was the Boxer Rebellion of 1900, when anti-foreigner, pro-Emperor peasant rebels attempted to drive out foreigners and foreign influence. Since there weren't actually that many foreigners in China, their victims were primarily Chinese Christians. Anti-Christian sentiment faded during the Nationalist period, as President Sun Yat-Sen was a (Congregationalist) Christian, as was his successor Chiang Kai-Shek. When the Communists took over, a group of Christian leaders met with Zhou Enlai to reassure him that Christianity was not a threat to the Communist regime. Several organisations were set up to control Protestant Christianity. The Chinese Christian Association, which regulates it, and the Three-Self Patriotic Movement, which is the only Protestant church allowed in China. It's vaguely Lutheran, but has Patriotism as one of its core tenets (this isn't emphasised in services though), and emphasises the Biblical verses about obeying the government (again, this isn't something that comes up in services, but it is their official policy). Similar organisations were set up to regulate Buddhism, Orthodox Christianity, Catholicism and Islam (but I don't know so much about them). This allowed the government to control and regulate the priesthoods of these religions (they didn't allow the Pope to appoint bishops), ensuring that anti-government religious leaders couldn't become leaders. Of course, there were also many underground sects that refused to join the state-run religious organisations, for a variety of reasons. In addition, as the Communist party became the vehicle to gain power and influence in society, the requirement for Party members to sign a document saying they agreed with the principles of the Party (including that they do not believe in any God), ensured that few Christians (or Muslims) could gain power of influence. (I should say that there is a rumour amongst Chinese Christians that Zhou Enlai was secretly a Christian, but I haven't found any evidence to support this) Now, in the late 1960s, Mao, fearing a loss of power, instituted the Cultural Revolution, an attempt to re-establish his power by renewing Communist ideology's hold on the people. One of the main instruments of the Cultural Revolution were the Red Guards, gangs of radical young Communists committed to restoring Mao's vision of a true Communist China, and stamping out the "four olds": old customs, habits, culture and ideas. It wasn't very clear what this exactly incuded, but it was assumed by the Red Guards to include religion, viewed as a tool of the upper classes to control the workers, or a dangerous foreign influence. Thus, many priests and monks, as well as other fervent religious practicioners, were sent to camps or beaten, often to death. It's important to note that religion wasn't officially illegal, but by the late 60s the power in China was in the hands of the Red Guards and their leader Mao, not the police, who were told to ignore Red Guard activities. But it was de-facto illegal. It's also important to note that it wasn't Christians who bore the brunt of this, but Buddhists (especially Tibetan Buddhists), because of their greater numbers. Nevertheless, priests and monks of all religions faced hardship and death. Many ancient religious buildings were also destroyed: in my city, Harbin, the Cathedral of St Nicholas, which stood at the centre of the city, was smashed to pieces. Even Confucius' tomb was desecrated.

After the Cultural Revolution wound down, with Mao realising it had gotten out of hand, and then his death, and the later arrest of the Gang of Four who attempted to continue these policies, the new de-facto leader, Deng Xiao Ping, who'd been purged during the Cultural Revolution, attempted to undo the damage of the Cultural Revolution. Statements made by the government seemed broadly supportive of religious liberty, and in fact a clause was put in the constitution guarenteeing freedom of religious belief. However, Christians were soon disappointed to find that they were still required to be part of the government-controlled religious associations, foreign missionaries were banned from entering the country (it's important to note that foreign missionaries did enter, under the guise of teachers, but this was illegal). Religious leaders who spoke out against the government were arrested, sometimes on spurious charges (the same as other dissidents). However, as Christians had gone underground during the Cultural Revolution, many simply continued to practice Christianity in unregistered house churches (these are also referred to as family churches in China). Without oversight, and limited communication with other Christians, some of these developed rather unorthodox, even cult-like, beliefs or practices. Which of course made it easier for the government to portray them as dangerous. It should be said though, that while house churches were illegal, they were not generally a government priority as long as they were out of the way, as they did not threaten its power. Membership of a house church was unlikely to cause you problems unless you were a Communist party member. But the main issue of Christians being effectively barred from the Civil Service or other powerful positions, while the youth continued their Marxism classes (ramped up after Tiananmen Square) where they were told that religion is dangerous and backwards.

So, that brings us up to the 20-year rule. Christians facing regulation, restriction, and legalised discrimination, but only severe problems if they rose to prominence or were very public about it, with arrest much more likely than murder.

If you want to find out about the problems faced by Christians in China today, there's plenty of news articles about it.

There is a well documented history of Christians being persecuted in Japan beginning in the late 16th Century and continuing into the Edo Period (c. 1600-1857). The short version is that, in the 16th Century, first Portuguese Jesuits and then Dutch Franciscan missionaries made inroads into the population, targeting elites and common people respectively, largely in southern parts of Japan. Christianity enjoyed some support – or at least tolerance – from the Shogunate up until Oda Nobunaga's efforts to unify the state. He saw Christianity as an uncontrollable and potential restive force (all the more so as a "foreign" religion), and effectively outlawed it in 1557. There was a famous incident in 1597 when 26 Christians in Nagasaki were crucified.

Many of the 200,000 or so Christians remaining went into hiding, developing secretive rituals in caves and the like to keep their faith alive. In public they would have to denounce Christianity (along with everybody else) through the fumie ritual, in which people stomped on clay tablets printed with the likeness of Jesus – something the authorities (mistakenly) thought Christians would be incapable of doing.

And indeed, the punishments for those revealed to be Christian could be dire. Some, for instance, were hung upside-down to literally drown in a pool of human waste.

For more, check out Christianity in Early Modern Japan by Ikuo Higashibaba.