A large chunk of grout came off from the area that covers the joint between our bathtub and the wall tile (see pictures). As I prepare to fix it I would like to know what is the best method to repair it, ie should I use grout or caulk and if so any specific product that I should get to match the current color?
OK, grout should never have been used there in the first place because it will crack. I can see more cracking from your pictures. Your best bet is get as much of that grout out of there as you can as it will continue to crack and fall out. Once you get the grout out, get some quality silicone caulk and run a good bead of caulk around the tub. Make sure everything is dry before caulking.