Reddit mentions of Methods of Mathematical Physics, Vol. 1

I'll list books that I've encountered along with some brief comments about them.

• I think that Jost's book is exactly what you're looking for. The last chapters in particular are excellent.
  
• If you're interested in avoiding the (necessary) difficulties associated with jumping directly into higher dimensional problems, you should also check out Buttazzo, Giaquinta, and Hildebrandt's book about 1-dimensional variational problems. I read mostly the early chapters, and I found it to be down-to-earth without sacrificing rigor at all.
  
• If you're interested in direct methods, you can't go wrong with Dacorogna's book on the subject.
  
• Although I've never personally used it in my work, I have had occasion to browse Fonseca and Leoni's book; it's without a doubt self-contained, although I wouldn't say it starts out explaining things like you're five.
  
• If none of these grab you, Courant and Hilbert have some helpful worked examples and discussions (of course, this book is older than the 1960s).
  
  Hope this gets the ball rolling!