I downloaded SNES SPC700 player (the one most people use), and it came with a DLL file. Do I keep snesapu.dll in the same directory as the SPC player or do I put it somewhere else with other DLLs? (I.E. System32, Program files, etc.)
Yes, the same folder works. Putting it in System32 will also work, but why didn't you just try it as is, with it in the folder? Typically you should leave files where they are unless a readme tells you otherwise.