This guide shows how to connect up an SD card reader to a Ben Eater style 6502 homebrew computer and use it to read data from SDHC cards.
The tutorial ends with being able to read arbitrary sectors from the SDHC card. Beyond that, I've included library-style code that can read from a FAT32 filesystem, with some example programs. For more information on this library code, see FAT32 Library Documentation
I'm only covering SDHC cards. You pretty much can't buy older/smaller cards these days, and not bothering to support the older protocol simplifies the code quite a bit.