What a great question. What is common practice in my department is to develop a working relationship with a professor who has shared research interests. This need not be your major professor. Approaching that person can be a little hard but if you've taken a class with him or her in the past then consider asking them if they have any interest in taking a term paper you submitted for credit and developing it into a publishable article.
I'm assuming that you have "Directed Individual Study" credits (or the equivalent) in your department - basically a one-one-one class with a professor of your choosing on a topic of your choosing. Taking a DIS is an easy way to get class credit for the work you do toward publication. And if you have minimum course load requirements (i.e. 9 hours) then this is a nice way to buy time in your schedule to work on the paper.
When you start with a term paper a lot of the hard work is already done for you. You only need to refine the paper and during the course of their grading they've already identified the areas that require improvement. They can also anticipate potentially damning reviewer criticisms so you can address them beforehand. They have a good handle on the submission process and can usually gauge what publications are suitable venues for your article. In the event you receive an R and R your co-author can be extremely helpful in responding to reviewer criticisms and, for me at least, in identifying which criticisms require action/revision and which should be rebuffed (and how to do that politely but forcefully).
Starting out small is fine but if the article is great then there is no reason it couldn't be published in a top tier journal in your field. My approach has been to shoot high and don't be afraid of a editorial deflection. I have a R and R on an article in well-respected journal and this is after I received an editorial deflection from a less well-respected venue. The lesson here is that sometimes the article is better than you think it is. Sometimes successful publication relies as much on matching the article topic to the journal's interests as on submitting a well-written, thoughtful, and timely article.
You know, after closer reading of your question I see my response doesn't answer your question. Perhaps I can salvage my response here though. I suspect that reviewer status for peer reviewed publications is shut off to graduate students because we aren't quite peers to those submitting articles (no Ph.D. just yet). However, I have a few colleagues who have reviewed articles submitted for inclusion in a book. Their involvement usually stems from their relationship with the professor(s) who are the editors of the volume. So it may be a function of who you know which, I know, is not altogether the most helpful advice.