Shooting in RAW in colour will give you the best flexibility. If you shoot in grayscale then you're stuck with the camera's pre-determined grayscale conversion which is most likely just a matter of placing equal value on R, G and B, this tends to create very flat looking grayscale images. If you shoot in colour then you have the ability to convert to grayscale while controlling how the individual channels to tweak the grayscale image. If you're used to shooting B&W think of this as using different B&W filters but that you don't have to decide which one you want to use until you see the effect, and you can change your mind. Want to mimic using a red filter, use more red in the conversion (or tone down green and blue, or a mix of both), green filter would be the same with green, yellow would be a mix of red and green (or decrease blue).
As for which mode to work in, absolutely don't use B&W, that's literally just black and white with no grays. Other than that, a lot of it is just down to how you like to work.
I've only used Lightroom a little bit (trial version) but was very happy with it. It's significantly cheaper than Photoshop and is very focused on photographic use (Photoshop has a lot of additional tools for image manipulation and compositing, etc which you probably won't use). As I mentioned, there is a 30 day trial version so there's no harm in trying it out.