I don't guess you know what brand of grout you have. The grout in my
new house has color matched caulks. Go on line and look.

I recommend you use a silicon based caulk, not 100% silicone. Regular
caulk is porous and will stain after a couple years, the color will
change as the dirt gets impregnated in it. If you can get a good
steady run going with the caulk gun, don't even use your finger to
smooth it out. Leave the pretty little bead there. A bead of caulk
will rarely crack whereas when you smooth it out to a flat or concave
surface, it will have more of a tendency to crack, especially if its
outdoors.

The grout is not going to stick to the baseboard. I have the same
problem in my shower. It has very nice shiny large tiles and they
grouted the bottom where the fiberglas base and the tile meet, it's
already cracking after 1 year. I will be shopping for the caulk
myself.