crafty kitchen

In my crafty endeavors this week is fixing up my kitchen. The sink and tile needed a little loving, and since my landlord thinks I walk on water or something, I get to learn and do it myself. So, a trip to Home Despot and quite a few youtube videos later, I have begun work. I'm repairing the grout on the tile and doing something about the seal around the sink, which I haven't decided yet how I'll do.

Here's the tile before (actually a photo of the side I haven't done yet):

And after (photo of the side that's done):
Pretty, eh?

Here's the dealio with the sink: Someone used grout rather than caulk around the sink. It is now really grody to the max, and leaks water down to the undersink when water splashed, as so often happens in a sink. Now the grout is all cracky and moldy, etc. The general consensus so far is to tell the landlord to do it, but I feel up to the challenge.

Here's the question: What is the best way to remove the grout? Thoughts? Here's a photo for reference:(And, yes, this is AFTER scrubbing with bleach. It doesn't get any cleaner. Ew.)

HELP!!!