It generally does not matter who files first in a New York divorce case. The filing spouse does not get an advantage to "set the rules" of the divorce. New York courts apply principles that do not favor one party over the other.
In 2010, New York became the last state to get rid of its "at-fault" divorce requirement. Couples now can get a divorce based upon irreconcilable differences so long as their marriage was not "working" for a period of 6 months minimum. In other words, a spouse may obtain a divorce if he or she is "sick of" the other person for at least 6 months.
On the other hand, if the couple is already separated and living in different states, then there may be an advantage to filing first. Different states have different standards for awarding alimony, child support and attorney’s fees, if those states even award such. Also, it may be of great inconvenience to one party to have to travel back and forth to the forum.
However, because of differences in state law, a person filing in a different state can take advantage of laws favoring their situation. For example, child support payment guidelines may differ in a neighboring state. But there are residency requirements as to where you can file your case. Generally, one spouse must have resided in New York for more than 1 year to file in New York.
An experienced New York divorce lawyer can provide you more information and help you through your divorce process. An attorney can also represent you in court and help you gather evidence for trial.