I'm doing multivariable calculus and I'd love if someone could shed some light on things that confuse me.
When we did integrals of real functions with real variables, the $dx$ that was in every integral wasn't really explained. It basically only had syntactical meaning. Now, it seems like the semantic meaning is what I'm having trouble with.
What does $da$ where $a$ is any symbol actually mean? Sometimes it looks like it's the differential of a functions (if $a$ is a function), sometimes it's just a projector to a specific coordinate and sometimes it looks like it's just syntax and it's ignored.
For example, when we did integrals like $\int xdx+x^2ydy$ and looked for a functions that produced that differential form we treated $xdx$ as, "when we calculate the partial derivative over $x$ of function $f$ we get $x$" (might not be correct due to bad English translation) but then when we looked at the "change in angle form" we treated $dx$ and $dy$ as something we need to calculate.
Also, where does the $\partial$ symbol come into play? People usually read it as just $d$ so I'm wondering how related are those?
I know this seems like I'm asking many question but what I'd really like to know is some intuitive explanation of the whole concept. Maybe someone else that was in my shoes before but "gets it" now can provide some tips?
The $dx$ and such that you used in multivariable calculus were indeed just syntactical. They didn't have any meaning or significance except to indicate what variable was the variable of integration.
When you did line integrals, you probably did something like this:
$$\int x \, dx + x^2 y \, dy = \int x \frac{dx}{dt} + x^2 y \frac{dy}{dt} \, dt = \int (x,x^2 y) \cdot \left[ \frac{dx}{dt} , \frac{dy}{dt} \right] \, dt$$
The "velocity" vector that was written $(dx/dt, dy/dt)$ is really just a tangent vector to the curve, whose length is determined by the choice of parameter $t$ that describes the curve. Ultimately, and especially in differential forms, almost all integrals are described in this way: by choosing some number of parameters that minimally describe the manifold of integration. Each of these in turn generates a tangent vector, and several of these put together characterize the manifold locally: a surface has two coordinates, and those two coordinates have associated tangent vectors that, together, span a flat vector space.
You'll notice here that the tangent vector is "acted upon" by another vector field in the integral. In differential forms, that other vector field is instead looked upon as a form, and forms must eat vectors in order to reduce to scalars.
Integrals using differential forms almost always use the basis forms that digest tangent vectors as cleanly as possible. For instance, if $r(s,t)$ describes a surface, then $\partial r/\partial s$ and $\partial r/\partial t$ are tangent vectors. Writing a differential form in terms of $\mathrm ds$ and $\mathrm dt$ is convenient because
$$\mathrm ds(\partial r/\partial s) = 1, \quad \mathrm ds(\partial r/\partial t) = 0$$
And similarly for switching $t$ and $s$. In this way, you can take a differential form written in terms of $\mathrm ds, \mathrm dt$ and integrate them: they consume the tangent vectors $\partial r/\partial s, \partial r/\partial t$ in the process of writing the integral, annihilating each other, and leaving only the entirely syntatical $ds, dt$ behind, which merely denote the variables of integration.
This is a point on which many people who use differential forms get confused about. $\mathrm ds$, the basis one-form, should not be confused with $ds$, which merely denotes an integral with respect to $s$.
So when you see an integral in differential forms, there's actually a "shorthand" going on: what you see written like this...
$$\int f \, \mathrm ds \wedge \mathrm dt$$
It really means
$$\int f (\mathrm ds \wedge \mathrm dt)(\partial_s r, \partial_t r) \, ds \, dt = \int f \, ds \, dt$$
That this is conventional shouldn't be forgotten: there is no canonical reason why we should put $\partial_s r$ into the 2-form first and $\partial_t r$ into it second. This in itself reflects the orientation of the manifold that we are not always free to choose.