Debian and Ubuntu are popular these days, and fairly easy to deal with. There are so-called "LiveCD" versions if you want to try them out before installing.
As for MAME, xmame works decently. At least most of what I've done with it has worked. It's based off the regular MAME source but draws to various Linux options (X11, svgalib, SDL, or OpenGL).
Make sure you use supported hardware for things. In particular, getting ATi graphics hardware working with Linux is a pain (especially compared to nVidia) if you want hardware 3d acceleration for other purposes, and you also need to make sure you have a supported sound card (most are). nVidia and Intel devices have full hardware accleration support for 3d and 2d, while most ATi devices have full 2d and somewhat limited 3d (even using ATi's horrible driver). Getting full 3d with nVidia will require some extra configuration beyond the initial install to get the vendor's drivers installed, just like with Windows. You'll have full 2d out of the box, though.
Any USB HID device should work, as will PS2 keyboards and mice. Be aware that if you need a custom tool to program something like a keyboard encoder, it probably won't be available for Linux.
There is a standard library for talking "raw" to USB devices called libusb (also used on Windows via the libusb-win32 flavor for many things) in case you need it, but unless you're a C programmer that won't interest you.