Visas? attending school

Hello:
I am an American in Sydney now visiting and I want to know which type are available to me for mainland China? I would like an L type with multiple entry if possible. I overstayed by one day in September 2013 and when I applied for one to return to mainland China from Hong Kong a week or so later I was denied. I am currently considering attending a Chinese language course at Dong Hua University in a few months and I would like to know if I need to apply there first and then be admitted and THEN apply for another type. Or can I apply for one (of any kind) and then go to Shanghai and register for school once I am in Shanghai? Also, how much are these types?