Shower leaking out the grout
Truthfully it looks like the original tile was not done with decades of service in mind.

You have two options


You could simply remove the existing caulking and pull out any loose grout from the inside of the shower. I wouldn't worry about the caulking and grout on the outside of the shower. The water shouldn't be getting past the first layer of protection anyway. Clean it up well, dig at it, use a brush and a vacuum. Let any damp spots dry. Then re-caulk will some tub and tile caulking. Make sure you get a mildew resistant caulking.
This is the long term fix. It's expensive but if you are going to do it right, do it fully right. Remove everything. Replace it with a waterproofing system like Schluter or Wedi. Some of the tiles on the outside of the shower door are showing signs of water weeping behind them. You can stop it by following the method outlined above. But when you are ready use Wedi or Schluter. Either of those systems would have to be done by professionals or friends who are fairly handy.


Wedi shower kit

masterwholesale.com

Schluter shower kit

www.homedepot.com