First off - job hunting takes time and effort. You're going to have to figure out how much time you want to spend on the hunt vs. how much to spend on school, enjoying life, and personal projects. It's not binary - you can do a little job searching, without continuing a full-throttle search.
Taking the questions in reverse order:
Is there a danger of burning bridges by doing this?
Yes, quite frankly, there is. I realize that the world is a chancy place, and there have been sad stories in recent years of companies backing out of hiring very good candidates due to drastic changes in financial state between hiring period and graduation date. But there are as many cases of candidates backing out on companies, leaving companies scrambling for new candidates at the last minute.
The biggest burned bridge is calling the company after you accepted and saying you won't come because you got a better offer. Particularly when it comes many months after your acceptance. You leave a gap in the staffing plan (companies generally don't offer expecting much attrition) and you may have caused the company to expend money and time that they can't recover (background checks are the big one). The nature of the burned bridge has a lot to do with the size of the company and the vindictiveness of the individuals involved. As a minimum, realize that you probably can't get another offer at that company.
Is it a good idea to keep looking for a job after accepting an offer of employment?
I'd advise a passive, slow speed "keep your options open" type search.
In all honesty, if what you think you can get is a "much better offer" - it would have been more ethical not to accept the first offer. If what you're worried about is that the job will dissolve because the industry is likely to downsize or the company is in trouble, it's more justifiable to look.
If you're doing an all-out job search, you're more likely to get another offer, but you're also more likely to be noticed by your current future employer. Settings on job boards are really the least of my concern, because so many people forget to change their settings upon hiring that I doubt most HR people worry much about it. But if you are searching intelligently, then you've written a great cover letter and resume, and you'll stand out.
Standing out is great in the job market, but not great from the perspective of a casual search. Apply thoughtfully to jobs you think are really great (way better than the current offer) and be clear that you're being picky at the moment because you already have some great options in the works.