Refitting patio tiles - Tile Advice

[alig9]

I want to refit a couple of risers which have come loose on a travertine patio and also regrout the joints on the risers. I would appreciate any tips/uggestions on doing the work and the adhesive/grout to use, especially as the area where the tiles belong does get a lot of the drain off from the rain on the patio. I have attached photos of the back of the tiles, the area where it needs to be attached and the broken grout