Publishing undergraduate research
There are many undergraduate journals that would not be likely to reject your work as "not profound enough." Basically, if it's written while the author was an undergraduate, and contains anything novel at all (at the level one would expect of an undergraduate), then a journal can be found for it. This includes well-written expository accounts of existing texts, especially if they work out some more examples. The most famous undergrad math journal is Involve, which has higher standards than the others (more like what a grad student or professor might do). But there are plenty of others that accept papers that might not make it to Involve:
- List of undergrad math journals
- List of general undergrad journals
- Tons of resources for undergrad research
Like the rest of us, you should do the research first, and think later about where it can be published. I agree with Sam that this should be guided by your advisor. Don't be afraid to have a conversation with your advisor stating that you are hoping for a journal paper in one of these journals. That can guide how focused the research experience should be, and can guide how you write the thesis. It's not overly ambitious at all.
One last point: I don't think it makes sense to tether your perceptions of admission to grad school to whether or not you have a journal publication. For one thing, even after you finish the research, it'll take you weeks or months to write the paper. Then, there will be 6-12 months while the paper is being refereed. Then a back and forth with the referees. The point is: grad programs in math would be crazy to expect undergrads to already have publications before applying. However, if you have a good draft, you can share that when you apply. I think it would matter less than your recommendation letters (another reason to talk to your advisor often), GPA, and test scores. Having fun with it, as you say, is the best idea. Whether or not you prove new results, your advisor can relay your passion and depth of understanding to graduate programs, and that will matter much more than your publication record at this stage of your career. Good luck!
I'm adding this answer in response to Annie Lee's question, because it's too long to fit in a comment.
Publishing as a grad student should definitely be done in consultation with an advisor. Unlike publishing as an undergraduate, a grad student's first papers serve as their introduction to the experts in their field, and largely determine whether they get a good postdoc. For this reason, it's very wise to have papers on arxiv before hitting the job market, but in many fields it's ok if these papers haven't been published yet (as long as the advisor's letter certifies that the work is solid). It's important not to put out anything shoddy, because your reputation really matters at that stage of your career. There's also the danger of getting scooped, e.g., if your PhD thesis has two parts, and you post the first part to arxiv a full year before the second, someone else might come along and do the second part before you can. An advisor will help you determine how likely this is to happen, and the pros and cons of advertising your work before it's fully finished. That's an important conversation anyway, so you can determine how much to say at conferences. An advisor can also talk to other experts in the field to encourage them not to try to prove what you are trying to prove in your thesis.
All that said, I think it's very valuable for grad students to have an early introduction to the world of publishing, as I previously wrote here. If you've never submitted a paper, responded to a referee report, etc. this can be anxiety inducing to do during your first job, while trying to juggle a million other things. Also, having an actual publication shows postdocs that you have what it takes to see a project through to completion. I was fortunate to write a paper as a second year grad student, in a totally different area than my main research (homotopy theory), and this gave me both confidence to get me through the hard times during my PhD research, and experience in choosing a journal, choosing an editor, when to keep polishing and when to submit, etc. Pro tip: the time to submit is almost always sooner than you think, because some degree of polishing can be done after the referee report, and the referee will always have suggestions for things to change (so, sending something too perfect can lead the referee to suggest big changes instead of obvious small changes).
In order to help give grad students this important experience with early publications, I joined the editorial board of the Graduate Journal of Mathematics, as I wrote about here. If you work out some interesting result or new example, and publishing it separately won't harm your PhD thesis, please feel free to submit to our journal! I think it's the only one of its kind, unlike the plethora of undergrad journals I mentioned in my other answer. To submit, you just pick an editor on the editorial board and email the pdf.