We added bathroom tub surrounds and tile regrouting to our service list this summer, mostly because we kept being asked. It is one of those jobs where a few hundred dollars of work now prevents a wall rebuild later.
Start with what you are actually looking at. Grey shadowing along the grout is usually surface staining from soap and hard water, and it scrubs out. Black speckling that returns within a week of cleaning is growth feeding on moisture that has already gone behind the tile.
The joint where the tile meets the tub is caulk, not grout. It has to flex, because a filled tub drops a few millimetres under the weight of water. Grout in that joint will crack every single time, and a cracked joint quietly pours water into the wall cavity with every shower.
Our regrout is straightforward. Rake out the failed lines, vacuum the dust, float new grout across the field, let it cure, then remove the old bead at the tub and lay a fresh silicone joint tooled in one pass. Twenty four hours later it is ready to use.
When we will tell you not to bother: if the tile moves under thumb pressure or sounds hollow across a wide area, the board behind it is gone, and grout on top of soft board is money thrown away. We will say so before we start.