Is it advisable in my case to take a second bachelor's degree? I graduated from an unprestigious school that most people consider a diploma mill.  I finished a social science degree in the College of Arts and Sciences of the school, but I’m ashamed to tell other people about it because of the school’s bad reputation.   

There are only few professors in the College of Arts and Sciences, since the school, based on what I’ve heard, is facing financial difficulties, forcing it to make cutbacks on the number of employees. The problem is, these few professors only hold a master’s degree, and sometimes they have to teach minor subjects that are far from their field of specialization. In one instance, a psychology professor had to handle a minor subject in philosophy.  More so, major subjects in, say, psychology are taught almost exclusively by only one professor. In fact, I only had two professors for all the major subjects required in my undergraduate degree. But this isn't the worst, because other departments sometimes accept fresh graduates to teach minor subjects.  The school also has a high acceptance rate and lenient retention policies. Hence, a student can enroll for the next semester even when he failed most of his subjects in previous semesters.

The only reason I stayed in this school is because I was given a full scholarship. Now, I’m desperate to finish a new bachelor’s degree. Should I pursue graduate studies? Any advice?
In almost every circumstance, one bachelor degree is enough. Even if it's from a less-than-ideal program, as long as you can show that you performed well, it will at least count for that.

If you are really that concerned about how your school appears on your academic record, I suggest trying to enroll in a Masters program at a more reputable school. Performing well at this level will allow you to improve your understanding in your field (and potentially fill gaps that were overlooked in your evidently underwhelming bachelors program) and demonstrate your abilities.

If for some reason you are unable to obtain admission to a Masters program, there are always non-credit or non-degree-track courses out there that you can take to bolster your breadth and depth of understanding as well as your credentials. Then, if you're still not satisfied, you can always reapply to those graduate programs.

But moral of the story, a second bachelors is not worth the time and money. Even once you have it, it doesn't say half as much about you as a graduate degree would in the same area. Top performance at the graduate level will allay any concerns about your undergraduate academic experience.