I don't believe what I'm doing is especially active or popular (so hopefully someone else will respond with a better answer), but seeing as no one has answered yet, I'll just mention one of the things algebraists do: invent new algebras.
The process is very easy to describe. It may or may not result in something useful. Take a set $A$ and define a set $F$ of operations on $A$ (maps from $A^n$ into $A$, for various non-negative integer values of $n$). The set $A$ plus the operations $F$ is what we call an algebra, usually denoted $\mathbf{A} = \langle A, F\rangle$. The algebras you already know (e.g., groups, rings, modules) are examples.
In my work, I think about different ways to construct such algebras. Usually I work with finite algebras, often using computer software like GAP or the Universal Algebra Calculator to construct examples and study them. I look at the important features of the algebras and try to understand them better and make general statements about them.
To address your last question, there is the following open problem that I worked on as a graduate student: Given a finite lattice $L$, does there exist a finite algebra $\mathbf{A}$ (as described above) such that $L$ is the congruence lattice of $\mathbf{A}$. This question is at least 50 years old and quite important for our understanding of finite algebras. In 1980 it was discovered (by Palfy and Pudlak) to be equivalent to the following open problem about finite groups: given a finite lattice $L$, can we always find a finite group that has $L$ as an interval in its subgroup lattice? Imho, these are fun problems to work on.
I can highly recommend "A Book of Abstract Algebra", by Charles C. Pinter. You'll learn about groups, rings and fields. You will also learn enough Galois Theory to understand why polynomials of degree higher than $4$ are, in general, not solvable by radicals.
It is 'formal' in the sense that it is rigorous, but the author is also very good at explaining the intuition behind all ideas. It is much less dense than most Abstract Algebra books, and, in my opinion, and excellent introduction to the subject.
Furthermore, it isn't expensive and it contains solutions to numerous exercises. See the amazon page of the this book for more positive reviews. Again, highly recommended!
Added: once you finished this book, you're ready for more advanced treatments of abstract algebra. After Pinter's book, you could try "A First Course in Abstract Algebra" by John B. Fraleigh. After that one, a great option is "Abstract Algebra" by Dummit and Foote. This is quite an advanced textbook, but a good one nevertheless. Once you've worked your way through these books (I advise you not to just read through them, but actually soak up the information by doing the exercises and reading actively) you will have a strong basis of knowledge in abstract algebra. By then you can tackle more advanced topics.
Best Answer
I'm going to give a short, very simplified overview of something that I'm somewhat familiar with, though there are many other open streams in research that I don't have the experience to comment on.
Let $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ denote the absolute Galois group, i.e. the group of all field automorphisms $\overline{\mathbb{Q}}\to\overline{\mathbb{Q}}$ fixing the rationals. Equivalently, $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ is the inverse limit of Galois groups $\text{Gal}(L/\mathbb{Q})$ of finite Galois extensions $L/\mathbb{Q}$, so in a certain sense it is made up of all finite Galois groups over $\mathbb Q$.
Perhaps the most well-known open problem in Galois theory is
An approach to this problem is through the famous Langlands program. A different approach was outlined by Grothendieck in his, also relatively well known, Esquisse d'un programme.
There Grothendieck notes that $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ has a faithful action on the collection of graphs embedded on compact surfaces, which he calls dessins d'enfants (children's drawings) due to their apparent simplicity. If one can understand this action, then in principle one can represent the elements of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ as permutations of dessins d'enfants. Thus one of the main open problems of the theory of dessins d'enfants is to
Shortly after Grothendieck's Esquisse went into circulation, Drinfeld proved that $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ injects into the so-called Grothendieck-Teichmuller group, which has an explicit description in terms of generators and relations. Hence one other open problem is
These are very difficult problems which raise further questions still unresolved, for example how can one compute a dessin d'enfant efficiently. Furthermore, theoretical physicists are also interested in Galois theory: Drinfeld's introduction of the Grothendieck-Teichmuller group was motivated by mathematical physics, and dessin d'enfants have already appeared in physics under a different name, dimer models.