While Calibre would be useful for pushing your papers to your ereader, it would be a terrible for library maintenance and I would also vote +1 for using a real reference program.
Zotero is great as a free option but I wouldn't use it if I was anything other than a undergrad; personal opinion of course. I would recommend using
endnote as it is widely considered the standard in reference managers, at least in the scientific community. It integrates well with MS Word (Without making assumptions on what type of grad prog you're in, if you write papers with 30+ references, you'll wonder how you ever lived without this feature), ensures proper formatting, and most journal repositories support direct exporting.
Of course it's not free and that sucks. Assuming you're not morally flexible enough to find a free copy somewhere, keep in mind that endnote is almost considered a mandatory program (at least it is in my discipline). Most universities offer this program at a steep discount to students (It costs $25 at my school). However, many departments also have site licenses you can tap into and as it's also degree relevant, it's not too hard to tap your grant to spring for it.
Like the "cloud" aspect of zotero? Then have endnote store all your files in
dropbox, a program that installs a "folder" onto your computer(s) that's really just a gateway to a free cloud-based storage service. The previous link gets me extra 500mb if you decide to install, if you don't want me to get extra space or simply don't trust me, just head over
http://www.dropbox.com. I've moved all my school related data to my dropbox as it allows for seemless integration and filesharing between my home computer, my laptop, my work unix box, and if it's a "not-so-smart day" my android device.
Still somewhat paranoid about losing your meticulously maintained collection of papers? No problem, since dropbox is just a "folder" all you have to do back it up is hardlink that folder if you use *nix or a mac (I think macs can do it, don't own one though) or, if you use windows, have it set to backup that directory or install
Mozy (once again a reference link, click
http://www.mozy.com if you don't like me).
My personal setup is endnote and dropbox on all my computers. Endnote imports to a specific folder in dropbox. This allows all my .pdf papers and my endnote database to be consistent and accessible across all my platforms. Additionally, I have mozy installed on my main computer since I have stuff on there (tax returns, personal photos etc.) that I want to keep backed up anyways so I have it also keep my journals safe while it's at it.
Lastly, the new firefox 4.0 (and the 5.0 betas) are fantastic, I would question the sanity of whoever told you otherwise. If you don't want it to be your default browser, it's a simple matter of clicking 'no' and unchecking a box when it first runs.
Hope this helps