Whenever there is a change of direction, caulk should be used. So between the step and wall should be caulk. Between tiles should be grout. The proper way would be to use a grout removal tool and scrape out the old grout and redo the gap with fresh grout. The tool should run you around $10 and they have small tubs of pre-mixed grout you can use for small jobs like this. Overall, this is a 30-60 minute fix.
Edit: if that little triangular tile wants to move you might need to stick it back down. I honestly wouldn’t both with mixing up any mortar just for that one. Some Liquid Nails construction adhesive should be enough for it.