I know very little about this period of history, and lately I've become quite interested in it. I am assuming that Jews began arriving in Europe some time after the Diaspora, but beyond that I really know nothing. I'd like to know something about the history of anti-Jewish sentiment in Europe in particular. Was this always the way it was? I realize this question is very broad but I think I'd like to keep it that way for now, since I don't really even know enough to narrow it down yet.
Jews began living in Greece and Italy even in classical aand hellenistic times. That means that, by the time of Augustus, they had already been living in Europe for centuries in small or large communities. Yes, your question is very broad, because it wishes to cover the fate of a nation and several thousand individuals over a span of several centuries: their degree of integration varies from time to time, and from community to community. It's not like the Jews of antiquity arrived in Europe in a few massive waves like their descendands did on Ellis Island in the 19th and 20th centuries. They were regarded as one ethno-religious entity among manyand treated as such. Anti-Jewish sentiment wasn't probably any stronger than anti-Greek and anti-Roman sentiment in Judaea in those times.
Some time ago I asked a related question -- specifically regarding the origins of the Jewish population in Eastern Europe -- and received a really great reply which you may find interesting, as well as a bunch of good source material. The respondent goes over Jewish emigration to Europe in general, not solely to Eastern Europe.
Regarding anti-Jewish sentiment in Europe, it has always existed to greater or lesser extents. Because of Judaism's strict monotheism, Jews were often singled out as a people who would not acknowledge other culture's deities. A lack of cultural assimilation would often keep them segregated in their own communities. These things could breed distrust among the rest of the community.