Removing grout from perimeter of counter and backsplash?
Use a tool like this: http://www.homedepot.com/p/QEP-Handheld-Grout-Saw-for-Cleaning-Stripping-and-Removing-Grout-10012Q/100001353.  



The link includes a brief video.  If the surface on one side is wood or Formica, be careful because the carbide will scratch it.  Typically, removing grout between tiles can only abrade hidden edges, and it gets covered by new grout.  You might end up needing to touch up where the grout line on the cabinet was using a cotton swab or a piece of cardboard as a brush.  If the Formica ends up with some visible scratches in the grouted area, you can typically find color-matched caulk.

If caulking is an option where the different materials meet, just remove as much grout as needed so that the finished job is neat.  The caulk can be a thin cover layer over the grout if the surface is clean.