I think some mp3 players WILL allow direct recording from CD, but not most, and not the cheaper ones.
But if you have a laptop, then you *can* download mp3's to any mp3 player, generally provided it runs a version of windows, or the player specifically supports the Apple Mac.
Is the reason your work PC doesn't like mp3's a restriction laid down by the company, or is it just not configured to play them? If its not to do with company policy, you can download free software that will rip(convert) your cd's and create mp3s. Provided of course you have a usb port, and a cd drive, and are NOT running Windows NT (which doesn't support usb).
But seriously I wouldn't even think of buying an mp3 player unless I had my own PC, as changing tracks would be too much of a bind. Even if you find one of the ones that WILL allow cd to mp3 recording, that would mean real time transfer speeds whenever you wanted a change...ugh.