Native parsing is done from the SVG file format, and people often start with files in one of those three formats. However, the answer in general is “It depends on what’s in the file.”
The AxiDraw is a pen plotter. Its sole function is to _plot vector graphics_, tracing the paths in the files that you provide to it. EPS, PS, and PDF files that contain vector artwork can be imported by Inkscape and plotted. This is often as simple as using File > Import. (Various other tools, besides Inkscape, can be used if you prefer.)
However, just because you have an EPS, PS, or PDF file does not mean that you have a vector artwork file. EPS, PS, and PDF files may contain vectors, bitmaps, or both. I can take a photograph, and save it as a .png file, but I could also save it as EPS, PS, or PDF — but doing so still leaves it as a bitmap, not a vector graphic.
Please read the answer above about .ps and .eps files. You do not necessarily have to import into Inkscape– there are various ways to drive the AxiDraw including a low-level serial interface. Printing directly from within Illustrator is not supported. If you are starting with graphics in Illustrator, I’d suggest saving as SVG directly from Illustrator, rather than going through EPS, PS, or PDF.