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Legally it does not matter who files first--there is no difference as neither party obtains a benefit. Just the terms change--the person who filed is known as a 'petitioner', and the one who replies to the filing is known as a 'respondent'. About the only 'benefit' is the one who filed ends up forcing the issue to go forward but that is it.
If you file first, you pay the cost of filing. The petitioner pays. But if he files in a different state from you, then yes, you'd have to fly to wherever he filed as that would have jurisdiction. Now, if you are in different states, then filing first can be a huge benefit both based on location and more favorable laws, depending on situation.