Q. How come some eat milchig first night Shavuos, isn’t there a mitzvah of simcha the first night too, and you have to eat meat?

A. Darkei Teshuva (89: 19) writes that many eat one milk meal on Shavuos. Others follow the tradition of eating milchig on the first night of Shavuos while consuming meat dishes on the day after. Orchois Rabbeinu (2 p. 98) writes that this was the Steipler’s Gaon zt”l minhag.
Sharei Teshuva (529: 2) quotes different opinions whether there is a biblical mitzva of Simchas Yom Tov at night or only a rabbinical one.
Moadim Uzmanim (7: 114) argues that in principle due to the simcha obligation one should eat meat at night, but maintains that one seuda suffices on the biblical level and two are needed only rabbinically. Therefore he purposes that the simcha obligation may be complied with by eating fish. (See also Sha’agas Arye 65 for some similar arguments.)
Yosef Ometz adds that simcha is created by consuming the right and propitious food for the occasion, as the Talmud suggested that the four wine cups could have been enough for creating simcha on Pesach night. Mishnas Yaakov mentions that the change to a milk meal, may be to grant recognition that this is a special siyum seuda, celebrating the completion of the mitzva of Sefiras Haomer.
Horav Shlomo Miller’s Shlit’a mentioned that the minhag of Lakewood Yeshiva was to eat both nights milchig, others would eat only the second night. His opinion is that everyone should follow their family traditions.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a