My experience with guice is that a great way to learn it is just to pick a few different scenarios for something your company already has, think of a use-case for each one, and work on implementing them using guice principles.
So when I went to learn it I took an already existing REST server that I had been working on for my company. I decided not to greenfield it, since I find the only real way to learn a new toolkit is to use it on a real-worldish project, and we already had a unit test suite set up for that service, so I would know if I caused a regression. I set a series of goals for where I wanted to take the project with guice, the first three were roughly:
- Use guice to replace all of the various singletons, DAOs, etc. Essentially objects that are poster-children for dependency injection and where DI makes them easier to test. This meant I had to figure out how to get guice integrated with junit very early, and also provided an opportunity to abstract out some modules I wanted injected during test and some other modules I wanted injected when in production.
- Get as much of the application to show up using the Grapher as possible. This also had the advantage of getting most of it integrated via providers, modules, etc.
- Abstract out the logging we were doing in the entry-and-exit methods using AOP.
So at the end of this I had a great graph of how our classes were interacting, all of the unit tests were passing, and I had simplified and improved our logging facilities for the REST interface.
I had substantially longer than a day (and I also had a longer list, including getting the JPA persistence piece integrated), but those seem like reasonable areas to start with if you have a suitable project lying around (or can make one quickly, e.g., by creating a quick REST service to do something that might be useful or fun).
I do highly recommend starting with something that either has tests or that you can write some basic tests for quickly, since that will let you tweak things or completely replace components and ensure that everything is still working.
Hope that helps.