See my October 1996 Best Practices column for an overview of security and what to look for. There is no such thing as absolutely confidential information on a Web server. This is one of the reasons many companies go to a hosting service. But there should be reasonable protections for confidential information built into the server software. This is important if you're planning on doing electronic commerce.
Most Web servers collect statistics by the bushel.
Not all of them are useful and the hosting service may not be willing to provide all of them. Find out what they collect and what you can get. You should get reports on hits, uptime, problems, and so on at least once a month. Weekly is better and some services make the logs available on a daily basis. Most hosting services can generate custom reports to your specification.