One of the biggest advantages to filing first is knowing it is filed and the process has started. You will not be sitting around waiting for the other side or the constable to serve you with papers. More importantly, either side can ask for temporary orders pertaining to the residence (who is moving out if you still live together) and either side can ask for temporary support for themselves and/or the children and in this proceeding, the party who asks for the temporary orders gets to go first. As petitioner, if you need or want temporary orders, you can ask first by asking in the Original Petition.
Going first at the hearing has a very distinct advantage: besides the obvious that you get the first word (in other words, you get to make the first impression on the Court) you and your attorney will get the control the order of witnesses at trial. While it is common for the Petitioner to testify first, you do not have to follow tradition. Your lawyer can call any witness, even your spouse (the Respondent) as the first witness. This means your experts, if any, don’t sit in hall charging you for their time, friends who are there to testify for you can plan their time and not be stuck in the courthouse all day, etc.
I love to call one of Respondent´s witnesses up front, sometimes because I know the witness will say something to help both sides and I want them to help my client first, sometimes because I want the Respondent to see me cross examine someone, so he or she will start to worry (this assumes I have information and plan to play rough with the witness) and sometimes it is to simply set out a simple fact that I then blow up into a big deal with a second witness. I also like to call the Respondent when I think he or she is not ready or expecting it. There is a sadistic pleasure in seeing shock on a witnesses face as he looks at his lawyer and says “you said she would go first”. Since the Petitioner gets to control the order of witnesses at trial, it is almost always better to be the Petitioner in a contested divorce or child custody case. Yes, Respondent can call witnesses in any order they choose, but the Respondent calls witnesses after the Petitioner is done; and why call someone who has already testified?
The advantage of making your spouse file are simple: he or she has to pay the filing fee, he or she has to hire a lawyer first, and he or she must make the commitment to follow through. If there are very little issues, nothing to fight over except hurt feelings, letting the other side do all the heavy lifting is easy, who cares who goes first if there is nothing to fight over. The Respondent can usually get away with a simple answer and signing the Final Decree and thereby save a lot on attorneys fees and cost. The Petitioner in this case pays all the cost and has to take time off work, etc. But, as stated above, the advantage in my opinion goes to being first. If you and your spouse have no assets and nothing to fight over, then you should consider an uncontested divorce where both of you use one lawyer, and get it over with quick and painlessly as possible and no one gets to go first.