Reddit mentions of Modern Physics

The Feynman Lectures will do the job, can be pretty expensive but you can just look at the online version here.

One book that deals with classical through modern physics is Physics for Scientists and Engineers with Modern Physics by Serway & Jewett. To (re)learn intro physics, really any similar book will do, and you can always get help from online resources, of which there are many.

A good text for Modern Physics on its own is Kenneth Krane's Modern Physics. It has a lot of problems (few physics textbooks don't, and you won't learn physics easily without them) but it has none of the other superfluous things you mention.

As far as math goes, maybe try using Khan Academy or a similar resource up through precalculus. As far as calculus is concerned, I can recommend Stewart's Essential Calculus as a pretty comprehensive textbook which covers a pretty wide area. I can also highly recommend Paul's Online Math Notes to help you learn algebra through calculus and differential equations.

Real answers for real high school student interesting in the conventional path to a conventional first course in quantum physics. Much of this advice applies more to the American school system, as that's where I was educated.

You're right, the first job is to get to calculus. Khan Academy is a good place for that! It's a bit messy, but just follow the knowledge web they have set up until you get to the topic of limits, which is the front door to introductory calculus. Along the way you'll also learn algebra and geometry. As soon as you can and as soon as you're ready for it, try and take a proper calculus class in your school. If you're in the United States, try to take AP Calculus.

If possible, take a physics class at your high school. If it's a reasonably big school, they'll have an algebra-based physics class and may even offer a college-level physics course that uses calculus (if you're in the United States, this will be called AP Physics C). Take this as soon as possible! If it's not offered, you may be able to take the equivalent course at a nearby college before you leave high school.

If you've done all of this right, you should know how to calculate things called derivatives and integrals, manipulate things called sequences and series, and understand the the basic rules of mechanics (force, momentum, energy, etc) and the electric and magnetic fields phrased in terms of calculus. In the language of most American universities, you now know Physics I & II and Calculus I & II.

Physics-wise, the usual next step is to take a course on waves, vibrations, and oscillations (see this table or contents) and / or a survey course on modern physics (see this standard text).

Math-wise, the next step is to take classes usually called

• multivariable calculus / vector calculus
  
• ordinary differential equations
  
• linear algebra
  
  The simplest way to do this is just to take these courses in a college or university, but there are also great online resources. I can personally endorse MIT OCW and Paul's Online Notes. Many schools also offer surveys of applied math at this level (with names like "mathematical physics" or "mathematical methods") that cover the basics of partial differential equations, fourier series, and more linear algebra / multivariable calculus / ODEs. See this book by Boas for a good idea of the content.
  
  Once you know all of that, you're ready to ready a real textbook on quantum physics. Some of the usual standards for a physicists' first course are the books by Griffiths or Shankar.
  
  Edited for link formatting
  
  TL;DR To go the physicist route, learn the following through school if you can swing it, but independent learning is possible and good resources exist online:
  
  
• Algebra I and II
  
• Geometry
  
• Calculus I and II
  
  Then these three in any order:
  
  
• Multivariable calculus
  
• Ordinary differential equations
  
• Linear algebra
  
  Then this, if you're going the usual physicist route:
  
  
• Mathematical methods
  
  On the physics side, you should take
  
  
• Physics I (sometimes called 'mechanics') with calculus
  
• Physics II (sometimes called 'electromagnetism) with calculus
  
  and then one or both of
  
  
• Modern Physics (sometimes called Physics III)
  
• Vibrations, waves, and oscillations (also sometimes called Physics III)

As for modern physics texts any of these should be fine but I have only glanced at them: Thornton and Rex, Krane, Bernstein, Fishbane, and Gasiorowicz

I read through Taylor and Wheeler's Spacetime Physics and it is really good if you want a lot of conceptual discussion of special relativity, not as much mathematics involved but honestly the math doesn't get too gnarly in SR anyways so conceptual might be the better approach to the topic. Unfortunately it only goes over SR, and not any of the other modern topics.