If you're a beginner you really shouldn't mess with CMYK. It's a can of worms.
There is no "generic" CMYK. There are specific CMYK profiles that correspond to certain press conditions and paper stock. Quite a few different standards exist for this, and you need to know which. So the printer should tell you which specific CMYK profile to use. If they can't - go elsewhere because they don't know what they're doing.
Image > Mode just gives you whatever you have set as working CMYK. It's basically just a shortcut. The default, if you don't change anything, is US Web Coated SWOP v2, but you don't know if that's the right one. It's just there because there has to be some default.
A business card seems like a simple thing, but there are countless traps here. I'd recommend doing it in RGB, but understand that many colors you see on screen are not reproducible in offset print. Dark saturated blues is a classic. If you have the CMYK profile you can soft-proof and be prepared for it or design accordingly.
If an RGB file turned out "too dark", you have some problem in Photoshop's color settings. Don't change anything in color settings unless you know what you're doing! The defaults are safe settings. Yes, you can change working CMYK to the one actually used, but you don't need to. You set this profile elsewhere, and then it will override the working space.
It also goes without saying that to have any predictability at all, you must have a properly calibrated and profiled monitor.
After u image> mode: cmyk
simply: file> save as
that should save a copy of the document in the mode you last converted to
be carefult not to save over your original rgb master copy, u should preserve that
If you can, recreate your artwork using Illustrator. You are limiting the use of your artwork working in Photoshop ( not that you cannot, you're just better-off creating vector artwork for both the logo and the business card file(s) ).
Another item to watch is resolution. The other is colorspace, rendering intent, and .icc tag. You're just beginning, so I am going to cut you some slack. Post again and tell us which RGB tag is assigned to the original logo file. Get Adobe's Print Publishing Guide where you will be introduced to the proper terminologies, workflows, file types, resolution, etc., etc.