I have been studying calculus for a long time, including multivariate calculus.
Now, suddenly, I'm reading about the concept of a "differential form", referring to $dx$ and $dy$ and so forth.
I've been seeing these symbols all the time in formulas like $\int f(x)dx$ and $dy=\frac {\delta f}{\delta x}dx + \frac {\delta f }{\delta y} dy$
My question is: is this term "differential form" an extension/advanced-version of the $dx$'s and $dy$'s that I've been working with, or have I been working with differential forms all this time without knowing it?
Yep, you've been working with differential forms this whole time without knowing it. In fact, the things which you would call vector field were actually differential forms (although vector fields are still a thing). Scalar functions are $0$-forms, the "vector fields" that you would integrate over curves (often written as $f\,dx+g\,dy+h\,dz$) were $1$-forms, the ones you integrated over surfaces were $2$-forms, and the scalar functions you integrated over volumes can be thought of as $3$-forms (recall that the integral of a $3$-form $f\, dx\wedge dy\wedge dz$ defined on a $3$-manifold in $\mathbb R^3$ is equal to the ordinary integral of $f$ over the volume).
Notice how the "vector fields" you integrated over curves are a different object than the "vector fields" you integrated over surfaces; the former is a $1$-form, the latter is a $2$-form. This would be more clear if we lived in a four-dimensional universe, for in that case, a $1$-form would have four components whereas a $2$-form would have ${4\choose 2}=6$ components.
Not only that, but the gradient, curl, and divergence are all special cases of the exterior derivative. For example, let $\omega = f\, dx\wedge dy+g\,dy\wedge dz+h\, dz\wedge dx$ be a $2$-form. Then $$d\omega = df\wedge dx\wedge dy+dg\wedge dy\wedge dz+dh\wedge dz\wedge dx$$ $$= \left(\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}+\frac{\partial h}{\partial z} \right)dx\wedge dy\wedge dz$$ If you carry out the computation, the exterior derivative of a $1$-form gives the formula for the "curl."
You may also recall that the curl of the gradient of a scalar function is zero, as is the divergence of the curl of a "vector field". More generally, when applied to a form, the exterior derivative applied twice gives zero.