Shower grout or alternative
Any type of filler other than grout shouldn't be used to fill the joints between tiles. You will need an approved cement-type grout either sanded or not. Sanded is called for when grout joints are wider than 1/8- 3/16 inches.
You can also use a two part epoxy-type grout which is more durable and mold resistant, but is much less forgiving to use.
Any caulk or silicone won't last very long in a wet environment and is difficult to apply to just the joints.
Re-grouting entails removing any cracked or damaged grout with a grout saw or similar tool. By hand it is a tedious and slow job. There are
power tools with attachments for grout removal that will shorten the work, but attention needs to be had in order to not damage the tile.
Once grout is removed from the affected area and the debris vacuumed out new grout can be troweled into the joints. It is easiest to color match the grout if there is grout left over from before. A similar color from the store may not match exactly.
Seal with an appropriate grout sealer when dried and before using the shower. (1-2 days).