Best project management software for non-tech project?

Hello,

I have been recently appointed to run a new program at a post-grad school as faculty. I will have to work with other faculty and staff and I need to properly organize tasks, goals, etc.

I have been working for few startups in the past and I am familiar with some project management tools, such as Basecamp, Pivotal Tracker, etc. but I have been away from that field for three years now.

What do you think would be a good tool to use at school to organize the new program? I need something easy to use (other faculties and staff do not have any previous experience with these tools) and the least invasive possible.

I will need to create goals, setup tasks for these goals, assign them to different people, monitor their status, share documents, keep discussions on track, etc.

The school does not have a Dropbox account. They are sharing docs and spreadsheet on the school's Google Apps accounts.

Thanks!