How does todays rapid access to information affect US relations with other countries?

This is completely a matter of opinion, and you could easily fill volumes in answering it. In my opinion, the overall impact is good. Why? Because many of the countries we deal with have government control of the media. The internet and other decentralized sources of information make it harder for those countries to mislead their people, including about the US. China is an obvious example. Of course, the Internet is itself rife with misinformation.

Additionally, many aspects of our culture are attractive to lots of people in other countries. For example, music from the US often finds receptive audiences overseas. Of course, those audiences can often avoid compensating the artists for their work.