How to behave when dating a fellow PhD student in same department?
One possible thing to watch out for: in some departments, graduate students may work as teaching assistants for graduate courses. You should ask not to be assigned to TA courses which your significant other is taking (and he or she should do likewise), as this would represent a possible conflict of interest. Even if you can grade your SO's work objectively, there might be a perception of unfairness among other students. Likewise, if any other situation should arise where one of you might be in a position of authority over the other, you'd need to disclose the conflict of interest and be prepared to resolve it.
Otherwise, as Peter Jansson said, relationships between graduate students are very common and aren't likely to cause academic concern. Just stay professional about it (e.g. don't make out in the hallways).
As long as you and your partner aren't put into any "boss-subordinate" professional relationship, it is nobody else's business what you are doing. So, it is perfectly fine to be in the same research group, the same class, etc., as long as it is not a direct "teacher-student" relationship. In the latter case just check what the university rules are.
The only (but major) negative consequence for the job is that if your relationship gets serious, you'll get a two-body problem to solve. I've been trying to solve mine since 1995 with no really satisfactory results (at best I could rank the arrangements I had as "tolerable"). So, watch out!
Concerning written rules you need to find out if any exist in your department, university etc. you will certainly not be the first or the only couple in this "situation". Unwritten rules are perhaps what you need to care more about and then I am actually not so much thinking about rules as such. First off, people know more about your relationship than you think. Or perhaps more precisely, they think they know more, in other words rumours and gossip might spread. so the advice is to be open about it and think about it as two adult persons in a relationship. It is your life, it is not a secret and most importantly, it is natural.
Regarding your work, you need to act professionally (like any other should). You need to help each other prioritize your respective careers and make sure your relationship can function despite the pressures graduate school inevitably involves. So in my opinion, you should worry more about balancing work and relationship so that you both can spend the time necessary to succeed. this means making sure the relationship does not directly or indirectly affect others on a professional level.