Like the others have said, the pan might be leaking - or the tiles might need a complete replacement. Sadly, most tiles are just adhered to green waterproof drywall. However, they should be attached to cement backer board - but, builders cut corners - and want to save a few $$ - so, doing it right rarely happens.
But, if you insist on replacing the grout, Dremel has an attachment that you can use with a Dremel tool for removing the grout. This might be the quickest method. I've used other "manual" tools with diamond tips, etc. - However, they require a lot of energy to get the grout out - plus, if you slip with such a tool, you'll scratch your tile. Once you have applied new grout to the tile gaps, ensure you use Tile Labs' Teflon-based grout sealer. A 16 oz. bottle is only about $15.00. This stuff is guaranteed for 15 years - vs. a much shorter period for other grout sealers.
Best of luck !!