Symbols are just names. You have a name, as do many other people. Some people even have the same name as you ! So really, saying "let x be a real number" is the same as saying "let us consider an employee from company X" (a theoretical company with infinite employees). If you want x to be in a certain subset of the real numbers then it's adding "let us consider specifically the janitors who work at company X".

So when you see a letter representing an element, think of it as an individual, relate it to something from the "real" world.

In order to truly understand your theorems and properties, what you must do is redo the proofs. Not copying them, but actually thinking and trying to come up with the steps after one read at most. After a while you will learn how to play with the words (symbols) and concepts. For example, the claim "a function that is differentiable at a point is continuous at that point" is not something you should simply admit but actually should prove.

Even if your professor gives you a demonstration for their claim, having it or reading it is NOT enough.

Also, try drawing stuff. For example, if you're working with sets and all, try making coming up with something that helps you visual it. It doesn't have to be "accurate" (everything can be drawn in 2D if you're brave enough, even if you're working on a space of large dimension, possibly infinite).

And finally, do exercises. Don't go directly to hard ones, start by the most basic things (example : prove that in a metric space, an open ball is open, if you've studied metric spaces; prove that a close ball is closed). The basic exercises and questions exist for a reason : they're the first natural thing you might think of and they are a direct application of definitions and theorems.

Saving this. You gave some great advice here.

This is really helpful, thanks! Do you know where the best place is to get practice questions with solutions so I can check after if I did it right?

Glad you read it ! Sadly I can't give you anything regarding practice exercises :/ I only use sources in french for that...