Each one of us learn something in different ways. For me, in 90's, I've got a nice material to study in the old Dr. Dobb's Magazine (Michael Abrash and his Graphics Programming column and later on Zen of Graphics Programming and Zen of Code Optimization books -- I was inspired by those), but stated with old 8 bit processors (Z-80 and MOSTech 6502 [Apple II]).
I wrote some "books" on the subject [sorry... in portuguese], trying to teach assembly to the novice, again, inspired on Abrash way of writing... And, never trying to build an entire project in assembly. If you are familiar with C I can only offer an advice: Use assembly only if you are certain that your routine is way better that one implemented in C. This because C compilers are very good (but not perfect) in creating optimized code. Most of the time, way better than you'll do.
My tendency, nowadays, is to create a routine in C; take a look at the assembly listing created by the compiler; and decide if I could do better (again, most of the time, I think I can't). Alas, this technique is good to learn how to create good assembly code -- and to refine your own C code.
This warning to almost never do assembly and trust the compiler has a purpose: Assembly code is often non protable. This is true, even, among the various "families" os architectures available for the SAME "kind" of processors: Exemple... if you are creating code for an old 486 and use R?? registers or SSE, it will not work!
My comment on "thinking in terms of math", by debs3759, is that technique is great, for every kind of programming (assembly or not), but I prefer to fill my code with meaningful comments, instead (lern this with Abrash)... The thing is YOUR knowledge of math and programming will improve, maybe a lot, in 10 years. Taking a look at mathematical driven style in the future, with new knowledge and new way of doing things, can make your own code unreadable to yourself. In that sense, I think writing a program is like writing a book or trying to teach someone (yourself included) what you did there.
[]s
Fred