You've missed the solid month of labor that this will be, the professional kneepads you will need to purchase, the multi tool you will need in order to be able to remove the grout, and all of the grouting tools you will need in order to regrout. And the large plus sized bottle of Aleve that you will need. You've just described one of the hardest most awful detailed work you'll ever do. It's brutal.

Really, it might actually be cheaper to put in new tile DIY. It'll certainly be easier on your knees and back. You'll end up buying most of the same tools, but you'll have them on hand when you want to do another bath or kitchen. You could probably tile your average bathroom for less than $200 in materials if you choose a plain tile and just jazz it up a bit with the pattern you lay it in.

Thanks Hollysprings. I think you might be talking about regrouting, which we're not intending to do (it sounds grueling!). We just want to make the most of our current tile and grout, and replace the caulking.
The old caulk has been ripped out this morning, and I must say things are looking much better already! Now for some cleaning...
Thank you!

If you have such a bad mold issue, it is because your shower is not draining properly or is not waterproofed properly. It just wasn't built properly. You can clean the surface of the grout, but it will not be possible to get cleaner/mold killer to the base of the problem, which is under your grout and tile. Anything you do, will be a bandaid, at best. Sorry.

Aggressive cleaning may buy you a little time, but realize the longer you put this off, the worse your mold problem becomes in your walls and floor.

Alice is right. All of your labor is only a very temporary solution. You really need to do a quick plan to demo and create new. Read this forum about how to create a waterproof shower and get started!

I agree that it sounds like you need to rip out & start over, but for a temporary fix...Use a razor to remove the caulk. Spray down with Tilex or other bathroom cleaner made for mold/mildew. Bleach in this case is not a bad thing--you will need it to kill what is growing now or it will simply come right back. Follow the directions on the bottle to the 'T'. Once everything is nice & white again, rinse it down, squeegee the surfaces including the floor & allow the shower to dry at least 24 hours. Then recaulk with a silicone based caulk that is waterproof. Wait another 24 hours before using the shower. Sealing the grout may help. Another tip is to use a squeegee on the tile (both walls and floor) after every shower--standing water is the medium that allows the mold/mildew to grow. Leave the shower door/curtain open to allow the shower surfaces to fully dry after each shower. Good luck to you!!

Thanks for all the replies!
Yes, we can see that redoing this shower entirely needs to move up on the list! We'll have to start saving for that.
In the meantime, you nailed it, we have a drainage issue - after each shower (2 per day), water puddles on the floor without draining. Until now, we've done nothing about this, but did purchase a squeegee today. I hope squeegeeing daily will buy us a bit more time.
Thank you!