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.
I suggest reading
O’Neill, Barrett, Semi-Riemannian geometry. With applications to relativity, Pure and Applied Mathematics, 103. New York-London etc.: Academic Press. xiii, 468 p. (1983). ZBL0531.53051.
It is 100% rigorous and covers most of what you want to learn, including basics of Lie groups, smooth manifolds and Riemannian geometry. Afterwards, you will likely need to learn hyperbolic
geometry, Lie groups and their representations in greater detail, as well as general principal bundles and vector bundles, and connections/curvature on general bundles.
Two more references:
Hall, Brian, Lie groups, Lie algebras, and representations. An elementary introduction, Graduate Texts in Mathematics 222. Cham: Springer (ISBN 978-3-319-13466-6/hbk; 978-3-319-13467-3/ebook). xiii, 449 p. (2015). ZBL1316.22001.
Chapter 1 ("Geometry") of
Jost, Jürgen, Geometry and physics, Berlin: Springer (ISBN 978-3-642-00540-4/hbk; 978-3-642-00541-1/ebook). xiv, 217 p. (2009). ZBL1176.53001.
However, you need a real advisor to guide you.
Best Answer
I am an engineer, and need to use the "practical" side of these things on a daily basis. Assuming you have a strong grasp of 3D calculus and linear algebra, you probably want to start with a bit of each of the following:
Multilinear algebra. This is where, e.g. tensors appear. Key topics are:
Calculus on (finite-dimensional, real) Banach spaces, generalizing (some of) what you learned in multivariable calc. Key topics are:
These should at least get you more comfortable working with calculus on more sophisticated objects than functions from $\mathbb R^n \to \mathbb R$. Once you're there, you may be interested in doing calculus on non-linear spaces... e.g. how do you differentiate a function on a matrix group, or on the sphere? At that point you may want to dip your toes into differential geometry... Key topics would include pushforwards of maps between manifolds, integration of differential forms, affine connections, Lie groups and the exponential map, etc... But that's probably a ways away for now :)