How to caulking a bathtub with various size edge between wall and tub
Just my thoughts on this:


I see mold everywhere, even by the faucet
the first thing you need to do is figure out what is behind the tile.  If it is drywall you are absolutely wasting your money doing anything.
if it is not drywall the first thing I would do is give it a heavy heavy spray of anti-mold cleaner.  Maybe 2-3 doses.  Scrub away mold using as little water as you can (this is a whole different question and will take a lot of man power).  I would then either not use the tub for a couple weeks or put up plastic sheets around tub.  You can get some painters plastic and tape it 5 feet high and your walls will stay dry.
if after a couple of weeks (at minimum) you still don't see the mold come back then we can talk about grout and caulk.  If the mold grows back either take down tile or keep plastic sheets up.  Either your mold is out of control or you have a leak behind wall.
if no mold you can grout and caulk.  Let's start with the gap by the tub.  This is too big to caulk for sure.  There is nothing wrong with grout here.  Do not install trim or other tiles at the bottom, that introduces more problems.  What happens if water gets in this first grout line?  It sits on tub, no big deal.  Your mold is coming from above.  After the mold is gone simply add more grout over this.  
for the rest you are are going to have to scrape out caulk and grout, regrout, then caulk and quite possibly put some sort of waterproofing on the whole area.  The water could also be entering through the tiles themselves.  If they are old ceramic these will let water through.
my final thought is just I would never worry about doing anything more than hitting it with some mold spray.  All the other time/money is a waste.  You can buy cheap tile for $100 and another $50 for grout/thinset.  Add another $50 for backer and you are at $200.  Thinking about the time and money you are going to spend I would just wait and do it right.