If you are just doing islands then you want to find an island group like the Cyclades, Northern Aegean, Dodocanese, etc., which will more than likely have ferry runs to other island in the same island group.
Lots of islands to choose from depending on your interests. You can get by with public transportation on many islands (but not all) and some times the schedules can be erratic or not work with your timing.
The mainland, especially the Peloponnese could take up all your time since it's filled with history, ancient and continuing up to modern times. It's where Greece began it's war of independence from the Ottoman Empire (Turkey). It would fill your entire stay in Greece. While you can do the Peloponnese by bus it will take a lot of planning and flexibility as schedule tend to be geared more to school and/or work hours. Some parts of the Peloponnese have better bus routes than other. A car would be my recommendation for the Peloponnese.
Athens deserves a minimum of two days with 3 better. There's more to Athens than just the tourist sites.
I like to book directly with an accommodation where you may get a better price, service and it's nice to get to "know" who you are dealing with.
I think a bit more research on your part to narrow down what you want to see and where you want to go.
Here's a website I use a lot for reference.
Lots of other websites so look around and don't overlook guidebooks like Lonely Planet or the Blue Guides to Greece