Why does the mishna say that child from the first marriage is a mamzer and also the child of the second marriage? If the first marriage was legitimate why is the child considered a mamzer? The child of the second is easy since the get was not legitimate.
Rashi (DH Tetzei) writes that the scenario is that she married on the basis of this invalid Get. Afterwards, she went back to the first husband (see Rashi DH veha'Vlad) and had a child from the first husband. According to Rashi, the Get is invalid according to Torah law, so she is really still married to the first man. Therefore, the child from the first husband is only a Mamzer of Rabbinical status. In reality, she was never married to the second husband but people might mistakenly think that the first Get was valid and that the second marriage was valid. If so, when she returns to the first husband without receiving a Get from the second man, people will assume that a married woman does not require a Get. This is why the child from the remarriage is a Mamzer mid'Rabanan.
Kol Tuv,
Dovid Bloom
The mishna does not say this nor does rthe gemora. This comes from rashi. It stoll does not answer the original question of why the mishna says first child is a mamzer. This is answee is fine if you accept the first marriage was invalid but this would not apply to first child who was born in a legal marriage.
The crucial point is that the first husband remarried her and the child was born from this remarriage (as Rashi DH veha'Vlad writes). Certainly a child from the very first marriage to the first husband is not a Mamzer because this was a perfectly legitimate marriage as you correctly write, but the problem occurs when the first husband reinstates his wife after she had married someone else in the meantime and did not receive a Get from the second husband.
Kol Tuv,
Dovid Bloom
Thank you. But again you need Rashi to clarify this since the mishna does not.
Yes; what would we do without Rashi?!
The Mirrer Rosh Yeshiva, Rav Beinish Finkel zt'l, used to say to people who learned Gemara without Rashi: "You are a Ga'on!" He meant that before Rashi came along, we had only the Ge'onim to help us understand the Gemara (and there are not so many Ge'onim still around!).
Mi'she'nichnas Adar Marbim b'Simchah,
Dovid Bloom