How would I go about getting a math PhD by the age of 25?

I have the will and the passion to want to get such a thing. I love math beyond all things. Being only 14, I (surprisingly) have found my passion and I know that a math PhD is my goal.

I’ve begun studying Linear Algebra and finished up with Multivariable Calculus in about a month. My dreams are set.

You have found your passion, and it is math. Your dreams are set. That's awesome (no, really. It is). Here's my advice, if you care to hear it: learn to play a musical instrument.

I don't mean that you shouldn't dedicate time to learning math. You should, and we’ll get to that. But given your drive, this isn't what you need to worry about. That'll happen. You do need to worry about not making your teenage years be about just this one thing.

When you grow up, several things may happen, some more likely than others. You may find that you have other interests, including things you never even tried, or knew existed. You may feel an urge to start a family, and you may need to learn to maneuver socially and romantically. You may feel burnt out by math. You may need to make a living and a suitable academic position may be unavailable. Who knows.

At 14, you may feel your life can be planned. It almost certainly cannot. Your job is to do the best you can to set yourself up for a fulfilling, happy life. Focusing on your passion is a good thing, but betting everything on a single outcome is almost certainly folly. There's more to life than math, and I say that as someone whose whole life is structured with math at its very core.

There's no particular need to earn a Ph.D. by age 25, specifically. You have a long life ahead of you. Don't optimize for speed. Earning a graduate degree by that time doesn't really force you to do anything crazy right now – you can comfortably finish you undergrad by 21 and complete a thesis in four years, it’s fine – but it doesn't matter. Maybe you'll already be married. Maybe you'll have had a job or two. Maybe you'll get your Ph.D. at 29 and find a position at a great university.

Don't worry about achieving X by age Y. Worry about learning, developing kindness and discipline, broadening your horizons, and (when you're ready), dating. I'm not kidding. I've seen teenagers spend years with math and little else, and there's a heavy price to pay.

Learning to play a musical instrument is a really good way to make progress along some of the non-math dimensions I've mentioned. It's not the only way, of course, but it's a really good one. Do it. Also, read books. Prose. Poetry. Study physics, and programming, and world history. Travel, if you can. Hike, bike or do some sport, any sport.

Now, as for the math itself: I don't know where you live and what resources you have available, but it'll be useful to find learning environments that emphasize math to some extent. Take advanced classes if you can. Take courses at a college nearby during high school, if you can. If at all possible find a tutor, a group of people to study with, a guide, a mentor. It's a wonderful journey but it's not easy to do it alone.

Make sure you spend time solving problems and proving things. Have someone knowledgeable review your proofs. Don't rely on your own feedback. Read books, and do the exercises. Reading alone is meaningless. Don't read the solutions unless you're really, totally and absolutely stuck, and if you are, ask for a hint first.

Knowing nothing else about you this is about as specific as I can get. Follow your dream and have fun, just don't forget to live while you're at it.