This is an excellent question.
Regarding your weakness in summation notation, that is one of the first things you need to address. I am a high school mathematics teacher, and I have a lengthy tutorial on summation notation, in PDF format, with exercises, that I give to my students. I have uploaded it to my ipernity account. It is in three pieces, because when I email it, using gmail, I am limited on file attachment size, and so have to email three times, once for each piece.
In order to download the three documents constituting the tutorial from my ipernity account, you would have to be a “pro” member of ipernity yourself. (Becoming an “ordinary” member of ipernity is free, which allows you to blog and send messages to other users, but you can’t upload/download documents.)
If you would like the material on summation notation, but don’t want to go the ipernity route, you can email me at my address given in my profile, and I will be glad to send it to you as email attachements.
The existence or non-existence of the “bridge” you ask about is a debatable point. One point of view, made by one of the other answerers, is that there is no such bridge, that you simply must keep building on what comes before. I take the view that there IS such a bridge, but that it is not an external object, but an internal process. You must become so adept at algebra, and closely related topics such as summation notation, that it is truly second nature for you. Reaching this point in skill constitutes the bridge. You can then move relatively easily into infinitary processes, of which calculus is the customary portal.
An analogy with building a campfire might be helpful. Correctly building a campfire involves three steps (after making sure you’re not building it under a tree!), namely, gather tinder, and light it, gather kindling, and add it to the fire, and then, only when a good blaze is going, add logs. The logs will then easily catch fire, and provide a nice long-lasting fire.
Using this analogy, it is easy to see the two kinds of mistakes that can occur:
Being happy with a kindling fire, that is, never adding the logs. The problem is, the fire will not last very long. (This is typically what happens in high school.)
Omitting the tinder/kindling steps and just dropping the logs onto the fireplace, and trying to light them with a match. Unless you have a lot of patience and stamina, you will simply give up and have no fire at all. (This is typically what happens in college –it’s the sink-or-swim approach.)
So, navigating that transition between the high school approach and the college approach is pretty much up to you, and will inevitably involve sitting quietly in a room. As Blaise Pascal said, “All the trouble in the world is due to the inability of a man to sit quietly in a room.”
Regarding specific books, besides what others have mentioned to you, I would like to recommend “Men of Mathematics” by Eric Temple Bell. Even though it’s been criticizd as not being completely accurate historically, it’s a great read, indeed, I daresay, pretty much an item of “required reading” for any beginner seriously interested in mathematics.
Also, the book “What is Mathematics?”, by Courant and Robbins, is something of a classic. I would suggest that it is pretty much required of any beginner seriously interested in mathematics to have held this book in their hands for at least thirty minutes, leafing through it:)
Also, addressing your concern about “vocabulary”, do you have a copy of a mathematics dictionary? The “Penguin Dictionary of Mathematics”, edited by David Nelson, is the one I recommend to my high school students.
Regarding study technique, there is some excellent advice here on MSE, in the form of an answer to a question. The question was “What are examples of mathematicians who dont [sic] take many notes?”, and the answer that I am referring to, which I upvoted, is that given by Paul Garrett. Here is the link:
What are examples of mathematicians who don't take many notes?
Also, here’s the link to a website you might want to consider:
http://www.mathreference.com/main.html
It “is essentially a self-paced tutorial/archive, written in English/html, that takes the reader through modern mathematics using modern techniques.”
You might want to check out the answers to the question here on MSE “How to effectively study math?” (which is where I pulled the above “self-paced” link from):
How to effectively study math?
And here's yet another MSE study advice link:
How to effectively and efficiently learn mathematics
As a parting thought, remember the story about the two mice who fell into a pitcher of cream. One mouse saw that the situation was hopeless, and so gave up swimming, and drowned. The other mouse could not see any way out either, but did not want to drown, and so kept on swimming furiously. And as it swam, its feet churned the cream, and gradually the cream turned into butter, creating a solid enough surface for the mouse to climb up on and escape from the pitcher.
So, the best of luck in your studies. Press on.
If you're interested in the subject, you should avoid getting trapped into thinking that mathematics consists only of those things you find in the curriculum of a university. That might mean you should look at some popularizations before deciding which things to try to learn. I read David Bergamini's Mathematics as a kid. It gives a different and far more truthful impression of the subject from what you'd get by doing the "bare minimum" in school. Not up to date but I still think it's worthwhile.
I like C. Stanley Ogilvy's Excursions in Geometry. It's amazing how much he can do with so little needed in prerequisite knowledge. Some of it is extraordinarily beautiful. (Follow the link and you'll see that one of the five-star reviews on amazon.com is mine.)
Some of the books published by the Mathematical Association of America are probably worth looking at.
At a more advanced level---say upper-division-undergraduate level--- (Part of the point of some of the comments above, is that you shouldn't only work at a more advanced level. But it's also necessary to do stuff at a more advanced level.) if you like discrete math, maybe Brualdi's textbook on combinatorics and Wilf's Generatingfunctionology. If you like stuff with lots of engineering and scientific applications, maybe Strang's linear algebra book. Very applied. (Here's the difference between "pure" and "applied" mathematicians: the former know about spectral decompositions of real symmetric matrices; the latter know about singular-value decompositions.) Regardless of whether you want "pure" or "applied" stuff, Dym & McKean's book on Fourier series and integrals can teach you something. (It's not very good for learing the analysis background; it's superb for reading about lots of examples of uses of Fourier theory.)
To be continued, possibly in a separate answer later, maybe........
Best Answer
I was in the same position as you, not too long ago. There are two ways (in my opinion) that intuition can be furthered.
One is in the sense of real analysis, which looks at continuity, differentiation, and integration from truly basic perspective. This is where you get deep intuition (in my opinion), because it reduces everything down to understanding distances between points and functions, which are very basic geometric objects one can easily visualize There are essentially no prerequisites beyond what you already know in terms of calculus. The book Introduction to Analysis by Rosenlicht might a good way to get this understanding, if you think this "atomic" outlook is the way to go.
Unfortunately, real analysis is rather extraordinarily tedious to read, and it is easy to be bogged down in definitions and minutiae. Also, you do not have much time before the exams to battle through an analysis book, in all likelihood.
So my suggestion instead is to do some interesting and difficult practice problems that do not involve rote application of formulas. For me, I was very motivated to learn the proofs for some of the most beautiful formulas I could find. Using only high school level single variable calculus (often via Taylor series, which you should know for the exam anyway), it is possible to show: \begin{align*} e^{{ix}} &= \cos x+i\sin x \\ \displaystyle \sum _{n=1}^{\infty }{\frac {1}{n^{2}}} &= \frac{\pi^2}{6}\\ \int\limits_{-\infty }^{+\infty }e^{-x^{2}}\,\mathrm {d} x &= {\sqrt {\pi }} \end{align*} although some might dispute the rigour of the proofs. See here, here (Euler's approach), and here. I guarantee if you can understand those proofs from beginning to end, you will be much further along. I found the beauty of these quite motivating.
My other suggestion is to look at things geometrically and visually. Consider plane and space curves, and their derivatives even. Use WolframAlpha to visualize some examples. Plot say helices and helicoids with their tangent vectors and planes at various points to see why derivatives immediately create linear aproximations to functions. Geometric and intuitive understanding will probably be more useful for you to do physics than analysis anyway (until you reach quantum and GR at least).
Lastly, some advice: there is no way around memorization of some formulas. Even as you get farther in math, you will need to memorize theorems. Nobody wants to (or can) derive every theorem from scratch during an exam! So just buckle down and do it :)
Maybe also look at the AP physics C curriculum (which you may already be taking). If I remember correctly, it uses single variable calculus to solve problems, which I found very pleasant for learning. In fact, very basic astrophysics textbooks are likely already accessible to you! They will use low-level single variable calculus and provide some useful motivation. For instance, Astronomy: A physical perspective by Kutner is on that level I think!