In most divorces it does not matter if you serve the divorce summons or your spouse does. The courts do not give you more or less because you or your spouse filed. You are not automatically perceived as the victim or the wronged spouse because you filed first. You will incur the expense of filing and having your spouse served, but otherwise the court fees will not be much different with a few minor differences along the way.
The situations where it does make a difference are if you and your spouse live in two different counties, then the person who files first can choose the county, but even that may have a few exceptions based upon where the children primarily reside. Another advantage of filing first is that it stops the future accumulation of retirement assets because they are valued as of the date of filing of the divorce summons. Certain other assets such as a business are often valued as of the filing date, but there are exceptions in that general rule as well.
Once the summons is filed "automatic orders" are in effect for the Plaintiff and then apply to the Defendant once the Defendant is served. These orders prohibit canceling health insurance, prohibit taking money from retirement accounts and borrowing or putting new mortgages on marital property. The "automatic orders" have other provisions as well and must be followed.
Should your case go to trial, and most do not get tried, the Plaintiff has to proceed first and therefore your attorney will initially have more work to do to prepare for trial. However, I would never advise a client to wait with filing because of this aspect.
Ultimately, this decision should be discussed with your attorney who can guide you and evaluate the benefits of filing or not filing. Your attorney may advise you to file if you need to obtain a temporary order of child support,
custody, exclusive use of the home or vehicle.