The minhag is to eat something milchig the first day of Shavuos. The Rama writes that the reason is in commemoration of the korban of Shtei Halechem that they brought on Shavuos, being that we eat both a milchig and fleishig meal, therefore one must have 2 separate challos for the 2 meals, since it is forbidden to partake of the same bread both for milchig and fleishig, hence, the commemoration of shtei halechem.
One need not bentch between the milchig and fleishig meal (unless he ate hard cheese) rather rinsing his mouth and washing his hands is enough. The zohar on the other hand writes that on must not eat milchigs and fleishigs in the same meal together, to be mekayem the zohar you must bentch.
Another reason for this minhag is, after the Jewish nation received the Torah, they went back to their homes but only found Dairy to eat being that meat takes a long time to prepare (shchita, nikur, melicha, cook in new kosher utensils), therefore they ate dairy, and we commemorate that.
Some also have the minhag to eat milk and honey since the Torah is nimshal to them, as the pasuk in shir hashirim 4' 11' says " dvas v'chalv tachas leshoincha".
