I heard a few rumours that knowledge of differential geometry is crucial to really understand anything in algebraic geometry. I want to ask You, people I don't know IRL to give Your opinion. If You know also about Lie's groups and algebras, You can write also.
I know that commutative algebra and sth my faculty calls "algebraic methods of geometry and topology" are necessary, but what about DG? Is it "must have" or is it only making learning algebraic geometry easier? And why it is like that?
I don't know anything about both of these geometries at that, moment, I'm asking in case to plan 2020/2021 season of my studies.
This might be true, depending on what you mean by "really". I think there are roughly two ways that differential geometry influences algebraic geometry:
First of all, the definition of a scheme (a space which locally looks like an affine scheme) basically mimics the definition of a manifold (a space which locally looks like a Euclidean space). As a result, arguments from differential geometry that let you translate local results to global results are also seen (in a modified form) in algebraic geometry. While it's not necessary to know any differential geometry to understand these arguments, it certainly can't hurt to be familiar with the ideas.
Secondly, many constructions on schemes are designed to replicate constructions from differential geometry in the algebraic setting, and while it's possible to learn the relevant definitions by rote, it's extremely helpful to have an intuition for these concepts from differential geometry. For example, if you've never seen the proof that the concrete tangent spaces of a manifold embedded in Euclidean space agree with the abstract tangent spaces $(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, it will probably seem quite strange to define tangent spaces in this way in algebraic geometry. The same is true for differential forms, vector bundles, etc. While you could learn all of this from scratch in algebraic geometry, it will probably feel rather unmotivated (especially because most texts assume you have some intuition about the corresponding concepts in differential geometry).
With that said, not all of algebraic geometry uses these constructions. There's quite a lot you can do with just algebra! You should just be aware that geometric concepts are quite pervasive in algebraic geometry (hence the name of the field), so you'll probably encounter them in most directions you look.