Hi, L.. I know it's really frustrating to wait to hear about a job, particularly if it's something you really need or want and are anxious about. I waited months after my initial application for my current job--which I desperately needed--before I got a call for an interview.
So, did you actually have a first interview, or did you just fill out an application online or apply through the company's web site? If you didn't already speak to someone, it would be highly inappropriate to contact the HR rep. If you did in fact interview or speak directly to someone already, then you should be fine to contact that person either by letter (more formal than email) or by phone, thanking them for their time and letting them know that you are still very interested in the position. If at that point you learn you are not in the running, it's a good time to ask for feedback on how you might tailor your resume to better highlight your applicable experience.
If you only applied and haven't yet spoken to anyone, it may be that your experience did not match up and you won't get a call for an interview--in which case there is no obligation on the company's part to contact you and let you know you aren't in the running. Remember, though, that "after the 4th of July" could mean either immediately after or many weeks (possibly months) after, so don't give up hope entirely. Also remember that the job market is really tough right now in Michigan. Keep getting your resume out and always ask for feedback. Get your name and your job goals out to anyone who will listen. Find friends who have spouses or friends who work where you want to work. Contacts are your best resources when job-hunting. Read and re-read your resume, and get opinions on it. Good luck!