Answers

First, clean out the existing caulk the best you can with a utility knife, flat razor or a putty knife.  To get it extra clean and depending on what type of caulk was there before, you can use baking soda and a rag to remove hard water build up and caulk residue.  Once you've done that, I recommend GE Silicone 2 or Dap Kitchen and bath caulk.  (I like the small hand tubes, not the kind that goes in a gun.)  Water based caulks are easier to work with, but silicone has it's advantages.  To apply silicone, clean and dry the area, apply a small bead, then spray the area with soapy water.  This will keep the silicone from spreading up the edge of the splash or onto the counter.  Then wipe once with your finger and your done.  The waterbased caulk can be applied in a small bead and then cleaned up/smoothed with a wet rag.

We like to use a sanded caulk that matches the grout for that seam between the countertop and backsplash. The flexibility of the caulk allows it to expand and contract without cracking out like normal grout. It's not a forever fix and needs to be touched up occasionaly, but it's a much better solution than just grouting that joint.

The trick with caulking is to spray the wet caulking with Windex and then tool it with your finger.  For "rookies" you may want to use masking tape.  Here is a YouTube video that might help, too. https://www.youtube.com/watch?v=uPGKdwnHhaE