You should always design your logo in black in white first.
Since you are past that part, converting in Photoshop goes like this:
- Complete all of you editing before converting the original image.
- Save a backup copy as a PSD before converting. This step is so you can edit the original version of the image after the conversion.
- Flatten the file before converting it. The interaction of colors between layer blending modes change when the mode changes. However if your file has vector text layers you shouldn’t flatten. If flattening might create some undesirable effects.
- Choose Image → Mode and the mode you want from the submenu. Modes not available for the active image appear dimmed in the menu.
Images are flattened when converted to multichannel, bitmap, or indexed-color mode, because these modes do not support layers.
After converting you can adjust the curves, levels, and brightness and contrast.
If your logo is a vector format you can easily convert it to black and white in Illustrator.
Then you can adjust the intensity using the saturation settings: Edit → Edit Colors → Saturate.
For printing format it really depends on what the logo is being printed on but the usual standard is a PDF. In a pinch I would use a TIFF or a JPG with a DPI of 300 or more but a vector PDF is really the way to go.
Take note that your quality will be lower if your artwork is raster or you take your vector art and rasterize it in Photoshop.