Job Application Process- Need advice

I am in the process of applying for new graduate jobs. Once I submit my application, should I follow up? I feel like my applications are just going into a big pile and I want to follow up, but without seeming annoying or pushy. Is it typical to call and check on the status of your application? This is my first "real" job and I don't know how the job process works. If I should call, would I just ask for the nurse recruiter and what should I say? I just want the hospital to know I am extremely interested in the position...I keep getting those auto emails from the places I've applied for stating that "so and so job you've applied for has been filled". So confused! Any advice would be greatly appreciated!