What does "va'Ta'an Lahem Miryam" mean?
Rashi: It means that Miryam sang and the women sang after her.
Why does the Torah write "va'Ta'an Lahem Miryam" and not 'va'Tashir' (as it does about Moshe and the men [See Targum Yonasan])?
Others explain that that is why they took drums (See Ma'ayanah shel Torah), and one can add that that is why the women left the camp to sing the Shirah in the desert.
See also Oznayim la'Torah, who, based on the fact that the Pasuk speaks in the masculine ("va'Ta'an Lahem" explains that she was merely encouraging the men to sing and not singing with the women.
What exactly did Miryam sing with the women?
Rashi (citing the Mechilta): Miryam sang with the women what Moshe sang with the men.
Ramban: They sang the first Pasuk only. 1
Why does the Torah write "va'Ta'an Lahem Miryam" in the masculine?
Riva, citing a Midrash: The angels complained to Hashem - if the men [of Yisrael] sing before us, will the women before us?! Miryam answered them - sing first, and afterwards we will sing.
Refer to 15:21:2:1**.