Firstly, get it digitally printed if you can - it will be way cheaper. To my understanding, multiple shades of grey/black is one color - just a variation of that ink.
If you are getting one side of the card in 1 color - it would be 1/0, if your card is greyscale, I really don't think it would be 4/0 - simply because they would not have to put in any other colors of ink...
Again, try and go digital - cheaper, good and sharp quality. My shop does digital cards on nice heavy stock for like $>125 canadian (depending on project) (1000cards).
good luck, PM me with more direct questions if you want.