Reading anything beyond serial number seams partially specific to the card type you have. Without a logic analyzer you are making blind shots which I really don't recommend.
Even getting the basic UID is hard work: as I've said, a minimal port of the public domain interface library comes to several thousand lines (OK, so that's well-formatted and properly commented but even so).
The best aid for this sort of thing is having byte-level debugging traces, at least until the library port is done. Anything more sophisticated will definitely need a logic analyser with proper decoding facilities.
However I think the most important thing is sufficient experience to be able to read the manufacturer's datasheet, which describes subtleties that you won't appreciate from inspecting code.
-----
If all that is needed is to get the UID of an HF card or the ID of an LF card I suggest using a 7941W module. It's a bit more expensive but is simple to interface via a serial converter (FTDI clone etc.), has the advantage of supporting both frequency bands, and the sensitivity appears a bit better than that of generic RC522 modules (which in common with a lot of other really cheap stuff are notorious for slipshod manufacturing).
MarkMLl