Role of Testers in Agile

Question

I work in a team which has been doing the traditional waterfall method of development for many years. Recently, we've been told that future projects are going to be moving towards an agile (particularly Scrum) methodology. It so happens that my project will be one of the first, so we will essentially be guinea pigs for the next few months to iron out what it takes to make the transition.

The project itself is in a very early stage and we would usually be many months away from releasing anything to the testing team, but now we are going to be working directly with them up front. As a result, I'm concerned as to the role of the testers in such a project at this stage. I have several questions/concerns which hopefully some experienced agile developers could answer:

1. While a developer is coding a task, it is impossible for a tester to test it (it doesn't exist yet). What then is the role of a tester at this point
2. Is the tester now involved in unit testing? Is this done parallel to black box testing?
3. What does the tester do during a sprint where primarily infrastructural changes have been made, that may only be testable in unit testing?

How do the traditional test team members function in your agile project?