1. The Romans didn'tcare about the Jews, one way or the other. To them, Judaea was just one more province, though an important one, as it connected the breadbasket of the Nile and the Near East with Rome. The Judaeans, of course, didn't appreciate the occupation, and rebelled against the pagan Romans. The Romans, in typical Roman fashion, crushed the rebels.
2. After Christianity spread all over the place that Rome began to target it. In fact, Christianity wasn't the first popular messianic religion to show up in Rome from the East. There were other cults that worshipped sun gods who died for your sins. Many Romans wanted a religion that promised them something more than an eternity in Hades, just as many Jews must have wanted something more than an eternity beneath the earth, in Sheol. Life under the Empire wasn't always so great for the common man.
3. The earliest recorded use of Christian, outside of the Bible, was when Tacitus recorded Nero blaming them for the Great Fire of Rome. Nero then persecuted them like whoa, and bought up much burned out city to construct an enormous palace. Nero was a crazy son of a bitch, his persecution of Christians likely stemmed from his psychosis as much as anything else.
4. In 313, Constantine made Christianity legal, and is known for being the first Christian Roman Emperor. After the leaders became Christian, they began to persecute non-Christians.