What is the best and easiest way to fix these issues? I assume I need to scrape away the bad grout with a grout saw, and then re-grout. Is there an easy to use product I can use instead of actual grout? Do I need to go ahead and re-grout the entire shower? The third picture shows the state of most of the tile grout.
I have already tried cleaning the grout with vinegar, and then with bleach.
It looks like most of the mold is growing, not on the solid grout between tiles, but on the flexible sealant (presumably silicone) joining the tub to the tile walls and joining perpendicular walls to each other. (If the edge between the tub and the tile is actually grout rather than silicone, that's a whole different problem.)
I suspect that this happened because the sealant was tooled into an incorrect shape that allowed water to pool against it rather than flowing down into the tub. The silicone appears to be peeling or broken in places, which could also be letting water in. Either way, removing the mold won't solve the problem, since it will keep coming back until you fix the silicone.
So your main task will be to remove the existing sealant and replace it with new silicone.
Note: you should never grout vertical corners between tile walls. Since grout is brittle, it tends to slowly crack as the walls settle over time; you should always use something flexible to join perpendicular surfaces. Those joints should always be silicone. A lot of people get this wrong, to be fair.
I soak bleach into a shop towel and let it sit against the surface for 8 hours. This typically kills any mold. Generally I use this at the caulking seams. The grout lines that you show is likely easiest to give them a quick sand with a grout saw and then use a grout sealer to prevent it happening again.
It looks like someone already tried easiest where they replaced the grout in the corners with caulk. To one degree or another, that's contributing to the problem you're having now.
I'd dig out the caulk in the vertical corner and replace it with actual grout. The rest of the grout can be scrubbed up with some grout cleaner and look good as new. It's the caulk that seems to be growing mold, etc. and it should be replaced. Grout cleaner and bleach, etc. don't work on caulk as well as it does on actual grout.
While you're at it, and working on the tub, you might as well re-caulk the seam between the tub and tile as that seems to need some attention in places too.
