Lately in my Physics C class we have been doing differential equations, which my teacher has explained more than once to me. Yet for some reason I can still not grasp the concept, I think my problem lies in not understanding the process of solving a differential equation. So my question then becomes what is the general process for solving a differential equation of the first order? I am a high school senior and a physics major by heart, so I really need to understand this process before I go to college. I am in Calculus AB and any answers are appreciated.
"Yet for some reason I can still not grasp the concept": By definition there is a concept of solution of a differential equation; but unfortunately there is no general method for solving a general differential equation. There are a few important types occurring all the time, for which there is a wonderful theory that leads to standard recipes. Such types you should recognize when you meet an instance of one, and you should know how to handle them. In addition there is a zoo of various special equations which can be solved explicitly either, and there are catalogues of such equations, notably Kamke's encyclopaedia. And for the rest, there are numerical methods galore. In particular: Forget about the idea of "integrating factor". It is heuristically helpful only in very, very rare cases; but above all it does not provide any geometrical or physical insight into the problem at hand.
The most important things about (systems of) differential equations are the following: (i) They enable you to describe in precise mathematical terms your intuitive physical ideas about how a (physical, biological, oeconomical, $\ldots$) "system" you are studying is evolving in the next second. (ii) There is a mathematically proven "existence and uniqueness theorem" that guarantees a unique solution for all times $t\geq 0$ when you have thought long enough about your "system" – even if you are not able to write down this solution in an explicit formula.
There can be various methods possible depending on the type and form of the differential equations but assuming your point of education level, i don't think you will be dealing with partial differential equations.
As for others, this article/paper seems to cover all the major possibilities you would encounter.
There are some well-defined classes of differential equations, and some specific ones as well, for which analytic solutions or methods are known. In general, differential equations are solved by quite a variety of different methods. If you are familiar with the concept of implicit functions, then you are on your way to grasping a differential equation, since DE's are implicit equations for functions which also involve the rate of change of the function with respect to one or more of the variables which are arguments to that function. Ordinary differential equations (ODEs) are for functions of one variable, and partial differential equations (PDEs) for functions of more than one variable. More here.