Bathtub Tile Repair

I mentioned recently that my bathroom tub tiles were loose and needed repair.  And since I am a reasonably talented D-I-Yr, I was asked why I couldn't do it myself. 

Well, I've never done any tile work, and this sure isn't the place to start...

Here is the humiliating proof...

I am quite sure that the tile backing is utterly rotted.  It may need studs replaced.  And I know for sure I don't want to mess with grout...

This is one thing I don't feel remotely competent to fix.