I heard that Romans broke the smicha chain for one to be a proper rabbi in the old sense, and I heard that rabbis started after the second temple was destroyed. So i'm confused, the two things I heard sound contradictory to me
What is correct?
Perhaps the reason you heard that rabbis began after the Second Temple, is that that is when the term 'rabbi' is introduced. For example, earlier Sages listed chronologically in Avot (1:2-12) are not called rabbi. These include: Shim'on HaTsadik, Antignos Ish Sokho, Yossi ben Yoezer, Yossi ben Yohanan, Yehoshua ben Perachya, Nittai HaArbeli, Yehuda ben Tabbai, Shimon ben Shetah, Sh'maya, Avtalyon, Hillel, and Shammai.[i] The first one in the list with a 'rabbi' based title, is Rabban Gamliel who indeed lived in the first century, the century of the Second Temple's destruction.
Alternatively, perhaps what you heard was part of a general doubting of the existence of a rabbinic culture in Second Temple times, given the seeming explosion of rabbinic literature and activity at the end of the Second Temple Era.
The traditional approach eschews this approach, and portrays a line of rabbis (although not called rabbis) from Moses to the Sages of the Mishna. The first chapter of Avot (referenced above) is one rabbinic source regarding the transmission of the Torah knowledge through the Second Temple era.
This is consistent with the Talmud's story (Avodah Zara 8b) about R. Yehuda ben Bava continuing the line of formal ordination in the early second century CE.
The Pharisees cannot have emerged suddenly, full-blown in the Hasmonian period. Their theology and organization must have been in formation somewhat earlier. How much earlier, we cannot say.
The Hasmonian Period was the second century BCE. This is at least a couple of centuries before the destruction of the Second Temple.
[i] I am aware that in later editions scribes sometimes added rabbinic appellations to these figures, but it is clear that these are later additions.
The correct record of how the Romans attempted to break the chain of rabbinic ordination and how the rabbis had it reinstituted is brought in BT (A.Z. 8b). Soncino trans.:
Forty years before the Temple was destroyed did the Sanhedrin abandon [the Temple] and held its sittings in Hanuth... Has not Rab Judah said [the following] in the name of Rab: Verily that man, R. Judah b. Baba by name, be remembered for good, for were it not for him the laws of fine would have been forgotten in Israel? 'Forgotten'! Surely, they could be studied? — Nay, they would have been abolished; for the wicked Government of Rome issued a decree that he who ordains a Rabbi shall be slain, likewise he who is ordained shall be put to death, the town in which an ordination takes place shall be destroyed and the tehum in which the ordination is held shall be laid waste. What did R. Judah b. Baba do? He went and sat down between two mountains and between two large towns between two tehums, namely, between Usha and Shefar'am and there he ordained five elders: R. Meir, R. Judah [b. Il'ai]. R. Jose, R. Simeon and R. Eleazar b. Shammua (R. Awia adds also R. Nehemiah).