I am really sorry that you are in this situation.
Consult a few (usually three) lawyers to find one that is suitable to represent you. Have the lawyers serve her with divorce/separation papers asap. The divorce process takes a long time. If she should redeem herself somehow, the process can be suspended or halted.
While waiting for her being served, open a separate bank account for you and have all your deposits directed to the new account. Withdraw 50% of all other bank accounts (unless they are specifically assets of your wife) and transfer the funds to the new account. At the time she is served, cancel all of her credit cards. Tell her to use a debit card which is attached to her checking account. Do not support her while she is unfaithful. She should not be able to finance her affair with marital assets. Your assets should be used for yourself, your children and their needs. If she has assets, discuss with her how you will allocate household and general expenses.
Implement the 180 process. The 180 is counter-intuitive, but it has been proven to be the best strategy, whether your objective is reconciliation or divorce. She needs to decide either him or you. If she decides him, then you have been "fired" as her husband. Since you have been fired, you should no longer provide any support to her, no favors, no errands, no nothing. Your focus should be on you and your children. Only have contact with her with respect to the childrens' needs.
You need to distance yourself from her so that you can prepare for her to move out. Tell her to go to the OM. She will see that you will not tolerate this blatant disrespect. Do not demand anything from her; demands foster resentment and are counter-productive.
Do not fall into the trap of the "pick me" dance. You can not compete with the fantasy and new relationship excitement associated with an AP. If you try, you will only make yourself look weak. By entering into an affair she has forfeited your emotional support. Until she drops the emotional and physical relationship with the OM, you should not provide any support to her at all.
If she picks you, she should go totally No Contact with the OM. Sorry, but your kids can not be friends. She can not have an affair and still be your wife. She needs to choose.
Obviously, no sex with her. Get tested for STDs. No telling where he has been. Move her out of your bedroom. She cheated, she should leave the marital bed.
Disclose her affair to your family and her family, and to your friends. Affairs thrive in the darkness of secrecy. If you have been "fired" as her husband, you no longer owe her any confidentiality. You do not need to consult with her before you take any actions. She didn't consult with you before she began her affair.
Go buy a voice activated recorder and carry it with you at all times. You must protect yourself from false charges of domestic violence or abuse. You might think she would never file false charges, but you never thought she would cheat either.
Other posters will fill in what I may have missed.
Do not cry in front of her, do not show her your emotions. She is not there to comfort you when she is having an affair with OM. The goal is to show "indifference". She is not the woman who you loved, she is now someone you do not even know, else how could she be betraying you.
Before you allow her back into your life, you must be satisfied that she is remorseful for her betrayal and she has fixed what has been broken in herself that allows her to enter into an affair. You may have a participation in any problems that existed in the marriage, but she is 100% responsible for her decision to enter into an affair and to betray the marriage, you and her children.
You must display strength and dignity. But you do not need to be polite or agreeable. You are not a doormat. You should not be disrespected.
Do NOT agree to marriage counseling. The marriage does not have a problem. She does. You did not cheat and you were in the same marriage. Marriage counselors job is to save the marriage. They will do that by getting the BS to compromise and to "forgive and forget". You should do neither until she fixes herself through individual counseling. Perhaps you should have IC as well to help you through this sh*tshow.
Spend time with your children as much as possible. Do not include her. At this time, she is not worthy.
Just for laughs, ask her if "open marriage" applies to you as well. Ask her if you are free to date and have sex with other women. It might be interesting to gauge her reaction.
[This message edited by PassThis at 10:13 PM, February 12th (Wednesday)]