In a few months, I think I'll be done with self-studying Introduction to Abstract Algebra by Nicholson. This book covers things like basic group theory, polynomials, factorization in integral domains, the Sylow theorems, series of subgroups, Galois theory, etc.
I want to be able to do some basic undergraduate-accessible open problems shortly, so when I finish this book (I am currently on chapter $8$ of $11$), where will I be in terms of being able to read a recent paper in algebra? I know algebra is a vast field and there are many subtopics (Galois theory, lie theory, algebraic topology, etc.), but I just want a general answer, no subtopic in particular.
It depends on what kind of paper you're looking for. If you're talking about current research papers in the subject, the answer is unfortunately no; I'm not sure how accessible such a technical subject is even to mathematicians from different fields, and there's a massive amount of group theory between an introductory book and what modern researchers in the field work on. (Take a look at some of the work on classifying finite simple groups, for example, which is brutally difficult.)
But it sounds like you're interested in finding something you can do now with the group theory you've studied, rather than just using it as preparation for further math work. To that end, here are some recommendations:
You might have better luck with books, even more specialized books than general or introductory ones, rather than research papers. I don't know what areas of math you're interested in general, but try taking a look at Griess' "Twelve Sporadic Groups" or the first few chapters of Silverman's "The Arithmetic of Elliptic Curves." Hungerford's "Algebra" is also a more advanced treatment of group theory and commutative algebra, though there's probably a lot of overlap with what you already ready. If you want to get in to even more advanced but still general group theory, I'd recommend Robinson's "Course in the Theory of Groups." Some of its topics, such as group cohomology, may not be as clear or motivated without some further background, but it's a good reference for group theory beyond the standard introductory material you'd find in Artin, Hungerford, etc. I also particularly like the combinatorial group theory in Serre's "Trees" (which will make the concept of free groups, for example, make much more sense).
Even if you can't get much out of a paper in group theory itself, you may be able to get through the group theory that's present in other kinds of research papers. There's quite a lot you can do with introductory group theory in number theory, for example; and you probably have enough group theory background for a lot of papers in topology, even if the topology itself is quite involved. (But bear in mind that that only applies to some papers; number theory has some incredibly complicated algebra in it, and there are topological topics like the Hawaiian earrings and their higher-dimensional analogues that do require some group theory beyond the standard undergrad/grad sequence.)
Even if you can't get through a whole paper, you still may be able to get something out of it. My field is topology, for example, and the mildly recent paper resolving the Kervaire invariant problem has an introductory section that clearly explains the problem and the plan of attack for it, even if the proof itself is quite complicated.
Your mileage certainly may vary, but one of the hardest things for me in learning to do math research was triaging research problems to find problems that were hard enough to be worth solving but accessible enough that I could actually make progress on them. If you have a professor (I'm assuming from the post that you're an undergrad) who'd be willing to point you towards some relevant problems, that would probably be much better than trying to find them on your own.