If there is moisture still there, then mold and mildew will return. If the problem was not fixed, then it will return. Bathrooms are particularly bad in this respect, since moisture is a natural consequence of using water. I would expect to need to redo things like caulking every few years to fix this. It will not last forever.
Scrape/pull off the old silicon.
Apply a solution of bleach to kill the mold. Be careful when you apply it, as bleach can damage the clothes you are wearing. Keep the area ventilated well.
Let it dry. If you are in a hurry, a fan blowing on the area will help. A hair dryer can also speed that process considerably.
Apply new caulk in a smooth bead. I use my finger to spread it along the joint. Definitely use a high quality silicon caulk with a mold/mildew inhibitor. These are caulks that are designated for use in bathrooms.
Allow it to fully dry before you get it wet again. Give it a day or so. It will skin over in an hour or so, but make sure it has properly cured before use of the shower.
Expect to redo this process in some number of years. The exact time it will last depends on how well the area is ventilated.
Of course, there may be mold in the wall itself. This is difficult to deal with unless you tear the wall down to replace it completely, and even then water tends to attract the stuff. Mold spores are everywhere.
The tile and grout in an all tile shower are not waterproofing materials but long wearing easy cleaning finish ones. IF your shower was built properly it goes like this from top to bottom...
-The tile and grout laid in thin-set tile cement.
-A layer of portland cement or thin-set cement
-A thick rubber waterproofing membrane
-A layer of portland cement or thin-set cement
-Plywood subfloor
The two layers of cement with the waterproofing membrane sandwiched between them is what does all the work of carrying the water from your shower to the drain and keeping it out of the rest of your house. Hopefully this was done properly...It can be tough to know. If you're lucky and you somehow have easy access it is worth inspecting the subfloor and joists underneath the shower. If I inherited such a shower that was having problems I'd cut a small hole in the sheetrock or plaster from the ceiling beneath the shower if it was in the way and I'd have a look or I wouldn't be able to sleep at night. A leaky shower of this kind probably won't pour water all over the place (unless it's really bad...) but let it weep and drip slowly, silently doing damage. The only way to know for sure is to look. I think this should be done first before anything else so you know what you're working with before doing any repair work. If it's anything but dry under that shower you'll sadly want to be redoing the whole thing...or leave it and never use it again...
If it is nice and dry, you should be good to go to replace the grout. It's tough to say why grout fails, as many factors are at work: the quality of the grout, the consistency it was mixed to, the humidity and temperature on the day it was laid in, how well it was laid in, the mineral content and pH of the shower water, and the list goes on and on.
The surest way to repair grout is to replace it. This involves scraping it out by hand first. There are different shaped diamond tipped hand tools for just this operation that ease the process a bit. You want to get absolutely all of the crumbly stuff out of there, and then even sound grout you'll want to gouge out so it's at least half the thickness of the thickness of the tile, and more removed is better. A strong vacuum is a handy tool to have for this process so you can keep the area you're working on free of dust and see what you're doing. Once you're satisfied you can simply mix up some brand new grout and lay it in there. A fine grit grout should be used.
Though there is no complaint of loose or broken tiles in your question, another possibility with these showers is that the structural system (the joists and subfloor) are not doing their job properly and letting the tile and everything beneath them to the subfloor flex, causing cracks. I've seen a shower like this built atop a plywood subfloor whose joints did not land on floor joists (...) so the whole thing was unstable and had to be ripped out and redone after stabilizing the subfloor.
Good luck. These showers can be monsters...hopefully yours was built well and just needs the cosmetic refinishing of the grout.
Best Answer
I have used vinegar many times to kill mold. Spray it on the area where mold appears and let is sit for 15 minutes, then rinse off. Bleach kills surface mold but doesn't penetrate below the surface to kill the "root".
You'll want to remove all the old caulking and then clean the area. Fortunately, vinegar works to remove old caulk, so does alcohol. After the area has completely dried, get a quality silicone caulk and apply it to the seam. let it fully cure, about 24 hours.