Although black grout could look great with what you describe, it is an absolute nightmare due to the extreme nature of the black pigment. A medium Grey will probably be your best bet all things considered.
There are plenty of sanded, and un-sanded Grey colors available, just make sure that they have both in your color choice. You'll be using a sanded grout where the joints are more than 1/8" wide.
Now the other part of the question: Grout first.
Then you'll need to find out whether or not there is a matching caulking. However, even if there is, I think you'll be better off in the long run going over the grout in the vertical and horizontal corners of the shower, and also around the perimeter of the bathroom with caulking over the grout.
Knowing what I know now about Lexel clear caulking, it seems to be the best choice in terms of longevity/durability, but also being able to go over it if needed without removing the old stuff.
It's really crystal clear and although it will make the grout appear wet and shiny in those areas, it is a small price to pay for an otherwise waterproof seal that will last.