I got a question from Colin M. recently during our study of Romans 1. If Paul was the apostle to the Gentiles, why did he spend so much time appealing to the Old Testament to support his positions (especially in light of the fact that he did not directly appeal to the OT in his speech on the Areopagus in Acts 17)? I've got a rudimentary answer but I need to do some fact checking first. I'll get back to you soon.
Some of my initial thoughts: where there were synagogues, they seemed to be bases of operations for the mission work of Paul; there were Jews in these congregations; the apostles looked to the OT as the authoritative revelation of God's word and Paul was just steeped in the narrative and teachings of the Old Testament and where ever he went he would use what he knew and believed to be authoritative to base his teaching in.
More thoughts: Judaism was a legal religion with holy writings that some Gentiles would have been familiar with; some of the initial Gentile converts were God-fearers (that is, Gentiles who had a strong affinity to the Jewish religion) and the people of that day were more open to religious matters, it wasn't compartmentalized like it is here in western culture.
I may do some more research and post again on this.