What can a student do regarding conflicting meeting times for desired courses? As it currently stands, my department offers two necessary courses at the same time every spring. One of them I must take before I am a senior, and another I must take before I graduate (I will soon be starting my third year). So I must take the former next spring, and I will have to wait until the spring of my senior year to take the latter. The problem is that the latter is a course that is very relevant to what I wish to study in graduate school. I'd hate to apply to graduate schools before having taken such a necessary course in my desired field (I feel it would weaken my application), and at the same time, it bars me from taking more advanced courses in the field before I graduate.

Of course, these courses are always taught at the same time by established professors, so I'd hate to ask the department to change all of that for one student. Is there any course of action I can take? Or would this be regarded negatively?

I hope that this is not off topic (merely because I am an undergraduate), as perhaps the desired rescheduling of course times is something that would be relevant to graduate students as well.
This is an issue that can also impact graduate students, so I do think it's appropriate for this group.  

Unfortunately, once a class hour schedule has been published for the next semester, and particularly after students have started to register for classes, it is usually not easy for a department to change class times.  Thus it's important to deal with such a scheduling conflict as soon as possible and preferably before students start registering for the semester in question.  

I would suggest that you send a polite email to the person in the department responsible for course scheduling (perhaps there is an undergraduate coordinator who does this, or perhaps the department chair supervises the scheduling) and explain your situation and desire to not have these two courses scheduled at conflicting times.  This is a reasonable request to make, but don't be surprised if your request doesn't result in a schedule change- in many departments courses are scheduled to satisfy the desires of the faculty rather than the needs of students.