Does notation ever become “easier”?

As others have pointed out, it gets much better if that's your first semester. But in my experience, there is not much relief between, say, years 2 and 4 of your studies. Sure, you get more mature, but the material gets more difficult too.

To address your question whether "more advanced mathematicians also struggle to extract the ideas from the notation", I'd like to quote V.I. Arnold, since I think it's exactly in the spirit of your frustration.

It is almost impossible for me to read contemporary mathematicians who, instead of saying "Petya washed his hands," write simply: "There is a $t_1<0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto {\rm Petya}(t_1)$ belongs to the set of dirty hands, and a $t_2$, $t_1<t_2 \leq 0$, such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.''

The trade-off is clear: without rigor math would've been quite a mess. But if rigor is the only way math gets communicated to someone, this person simply won't have time to get far in math.

The question you need to ask yourself is: "If a lecture contained only ideas and intuitions without rigorous formalities, would I be able to write formal proofs on my own?"

If the answer is "yes", then congratulations, you are very promising young mathematician. If it is not the case, you are in the same boat as most of us were when we began studying mathematics (you might still be promising young mathematician, though).

The thing is, formal language is a necessity when we want to clearly state our ideas, check if they are correct and share them with others. Sure, two experts in a field might not need to be very formal when communicating with each other and still understand perfectly what they mean and you listening on the conversation might not understand a single thing they said. If you were interested, you'd want them to tell you the appropriate formal definitions and theorems involved. You might want to know about particular details in some proof and how does one reach desired conclusions. This would not be possible without formal context. Then again, even two experts might find themselves in disagreement in which case they would go back to being formal to clarify things to their satisfaction. Lack of rigor can obscure subtle errors and we've all fallen prey to it at some point. Hence, it is essential for any mathematician to be able to read formal proofs and to write their own.

The beginning courses in mathematics ought to teach you rigor and formal language in order to prepare you for understanding advanced topics. You might struggle now, but when you grasp it, you will be very thankful for it. The goal of learning proofs will shift more to understanding ideas involved and not the formal language, but at that point you should be able to write down ideas formally and judge their correctness by yourself.

To be fair, I agree that some authors are overzealous with formal language while disregarding intuition and ideas. But, this is what the faculty is for, professors and teaching assistants are there to explain and clarify. When in doubt, you should ask for their help.

If its your first semester taking a proof based class, then its very typical that it takes a long time to absorb proofs. The more math classes you take, the more fluent you'll become in reading and understanding proof. It gets better!

In my experience, there is a trade-off between compact notation and lengthy prose in math texts. Sometimes mathematical texts try to introduce lots and lots of notation so that what you read is more like a condensed code to be processed by a computer rather than to be read by a human. At other times, some authors use too much text AND too much notation taking the reader safely through each step in a cumbersome manner where you as a reader lose track of the goal of and idea behind the whole thing.

Getting used to interpreting notation is of course always a good thing, and in my experience this happens when you both read and write mathematical notation yourself. Sometimes writing out the compact and complicated statement from the textbook will help you grasp the idea condensed in there. Condensing lengthy prose works equally well. This way you will keep getting better.

It is just like learning the alphabet - it takes time, at first it seems ridiculous and incomprehensible, even if you get the gist of the story you are trying to read, it's hard to learn the specifics of the language or story or whatever. But when you do get it, it's easy to read and write, and understand what is going on, and you can't imagine ever having not understood it - it's easy too you.

Practice makes perfect - the more you practice with it, the better you'll get, and eventually it'll be hard for you to imagine having not understood all of that notation. So keep reading proofs, writing proofs, and trying to swim through the quagmire of notation - it'll get better.

"...in mathematics you don't understand things. You just get used to them."
- John von Neumann

This Question and its Answers explaining the quote seem to incorporate most of the possible answers to your question.

It gets much easier to understand the notation. At a certain point you start to realize that you've seen all of the basics - the hardest notations you'll encounter are the ones introduced in the proof you're reading, and then the definition is right there for you to look at. You'll also find that understanding the topic better makes you understand the notation better - if you have a good enough sense for the situation, you can tell what the notation should be saying. You know you've started to hit this point when you start being able to correct typos without getting confused.

So my advice is to aim for an intuitive comprehension of the topic you're working with; start out by ignoring notations and just getting a feel for the situation. Once you get the hang of it, look at the notation again and see if it makes more sense.

The answer is that these things build one upon the next. It is pretty unusual encountering math books where the majority is presented in notation but you learn basics and work your way up, as with anything else. It becomes easier over time in other words. Trying to read a topology text without a foundations course and some analysis would be damn near impossible. So take heart in that it will get easier.

Others have already answered whether it gets easier during the course of your studies (it does). Suboptimal notation exists, and teachers who use suboptimal notation also exist. (Some teachers may even be deliberately trying to teach you how to chew through suboptiomal notation.)

Reading your question differently, there's also the question whether notation improves ("becomes easier") over historical time ("ever"). It does. Evolution works on ideas, including notations. But we have to be patient, it may take a century or two.

When an idea is first described, it is often done using existing notation. This makes sense, because the goal of the description is that readers understand it. New notation might discourage readers, putting the new idea at a disadvantage. Later, when the idea itself is widely understood, people may strive to streamline things, and switch to different or new, better suited notation.

Some scholars opt for a new notation immediately, from first publication, some successfully and some not so. For example, Hermann Grassmann used a new notation to introduce his ideas on multidimensional algebra. No one influential understood what he was writing (or publicly admitted to do so), so his ideas were not widely recognized. Clifford and Hamilton pursued similar ideas, with more success, possibly because they used different, more common notation (but probably more so because they were more influential to begin with).

A well-suited notation can definitely make it easier to reason about a subject. This of course applies not just to mathematics, but many subjects, for example physics, or programming. Putting a little effort in finding, adapting or crafting the "right" notation for the job is often worth it.