> In short, how can I become smarter?
Work thoughtfully and hard, and take nothing for granted. That's it.
> How can build a great foundation where I can understand the concepts intuitively?
You're lucky, because you're in college. It's the best place to do it.
Know the material. If you can't draw a picture in 1 minute that would get the idea across to your grandmother, you don't understand it. Every time it says "Clearly," you prove whatever comes next. You close the book and put it away, and come back the next day and do the same thing.
Do all the problems. especially the hard ones at the end. You never look at your notes or your book when doing them. That's the difference between getting an A on the exam, and being able to teach someone that topic five years later. Your study time (at least of fundamentals, which is most of university) should consist ~70% sitting in front of a knotty problem, with no books in sight. It's recommended to do mainly hard problems, and it's absolutely OK to take several days to do them. If you don't make progress for more than 20 minutes, stop and come back a few hours later or the next day. But make sure you come back!
Now, here's the key: you do all this before the lecture on that topic. When you show up to lecture, you already know it cold. Except that you'll have so many questions. Most of your questions will start with "Why." They will be the questions that the other kids will find annoying. If you want to be smart, you need to be that guy.
Teach. It's trite but true. Teach people who seem smarter than you but don't know about this topic yet. They'll help you understand what you don't know properly. Also, our retention of information processed in a live setting is roughly 9x that from just reading.
Be curious. Our brains are wired for stories rather than abstractions, and even technical fields have a narrative. Who came up with this algorithm? How or why did it follow logically from what came before? What came next? By learning these things (the best way is to chat about it during office hours with professors, or with other smart students), you'll also be getting an informal education in what smartness and creativity look like in the real world.
Focus. Do not do what's above for every class, or every topic. Do it for only one topic at a time, that you're really interested in, exclusively.
Form a team. This is probably the most difficult (at least for shy people like me), but the most rewarding. If you know some other people who are also serious, and can get together and work with them in the fashion above, you will all progress much, much faster. Try to have some and discipline: you meet twice a week for two hours and work like crazy, and then go to the pub afterward and wind down (just like a sports team). If you can find a grad student or prof who's interested in "coaching" you, that will help even more. This is what grad school has that college doesn't. Create grad school for yourself in college, and you'll get smart much faster.
Play. Creativity emerges most spontaneously during play. Play means pet projects, hackathons, competitions, etc. Research projects during the summer are perfect (choose them based on coolness and/or fun). And play in completely unrelated areas, too (music is fantastic). Non-academic recreation (e.g. lots of sex) is quite important.
> .. go to graduate school at Stanford or MIT
Question: why do you want to go to grad school at Stanford or MIT?
> I would just like to prove to myself that I have what it takes
That is probably not a good enough reason. It's easy to become enamored with the idea of a thing. Spending all day wrestling with deep problems, letting your mind course over the beautiful edifice of ideas erected by those who came before. Unless you actually enjoy doing spending most of your day and most of your nights alone working, usually in a library or cramped office - then, perhaps there are other forms of "smartness" worth pursuing.
It's a decision only you can make - but if it's just about pride, I'll save you a lot of trouble and tell you it's not worth it.
> I've been feeling pretty lost in what I've wanted to do with my life
Most people feel the same way. It will not stop you from living a worthwhile life. The majority of interesting events and true life experiences you have will be unplanned - so don't take "want to do with your life" too seriously.
Have you ever seen the movie "Yes man" (Jim Carrey)? Whatever happens, he just says yes. As far as I can tell (at the tender age of 24) this is a good strategy for a successful, rewarding life. It may sound at odds with the "Focus" paragraph above, but it's not. You focus to improve yourself - and you say "yes" when serendipity comes knocking, often disguised as drudgery.
Good luck :)