Oh man, been there, done that.
My advice:
1. You need to accept that your marriage is over.
Meaning, there are five steps to grieving and you need to get to the fifth step (acceptance). You don't have to hurry to get there, but you need to get there. It sounds like you are more than a little bit down this path, which is a positive. A divorce is a legal action/business decision. Detach yourself from the emotion of it all as best as you can.
2. Get a lawyer.
Now.
Getting a lawyer does not mean that your divorce will be combative. It means that you have a legal advocate through a lawsuit (a divorce is a lawsuit... one of you will literally sue the other for divorce). You need to know your legal rights and you need to work towards a reasonable resolution.
Right now, you are feeling sorry for yourself and you are overwhelmed by the thought of being a single father. You are potentially facing significant alimony payments and child support payments. You absolutely need an attorney to help you navigate these waters. I can't help you, at all, because you don't live in my State and laws vary from State to State. But, because you are the primary money-earning from your marriage, you need to know what your future financial status will be. For example, my state assumes that every adult can work a full-time job. My ex worked part time and she wanted to use her part time salary for the purposes of computing child support and alimony -- this was such a huge sticking point that we literally went to trial over that one issue.
3. Be prepared for an ugly divorce.
My ex said that she wanted a quick, amicable divorce too. She then proceeded to do everything in her power to ruin me financially and emotionally. I hope that doesn't happen to you, but the same personality traits of a person who is a repeat cheater... are the same personality traits of a person who goes bonkers during a divorce.
I'll sum of my divorce quickly: We had 10 of something (it doesn't matter what specifically). The law says that she should get 5 and I should get 5. She asked for 7 and I asked for 5 and then she accused me of being unwilling to compromise because I wouldn't "compromise" and give her 6. Be ready for this type of stuff.
Hopefully, this doesn't happen to you. But, put some effort into making sure that you have a thick skin going forward.
4. Be polite and kind to your soon-to-be-ex at all times.
Any hint of domestic violence can be used against you going forward. Do not instigate anything. Be polite. Smile a lot. Keep conversation to a minimum. Don't listen to any of her legal advice or crazy theories. Stop worrying about whether her current thing is an affair or not. She's now your ex and she can sleep with as many people as she wants (and for the record, if she isn't cheating with this person... she's probably cheating with a different person that you don't know about. Exit affairs are thing, yo).
I recommend that you keep verbal conversations to a minimum. Try to communicate via text message and/or emails. Keep all communications.
A few stories from my divorce...
We did in-home separation for almost a year. During this time, my ex-wife accused me of blocking her in a room on three separate occasions... and she absolutely was taught by her attorney to instigate these incidents. On the first two occasions, we were having a somewhat normal conversation while I was casually standing in a doorway, leaning on the door frame. She then proceeded to yell "move!" at me and when I did not immediately move, she accused me of keeping her in a room against her will (which is illegal). Of course, I proceeded to let her through after she clarified that she wanted to leave the room (i.e., "move!" is very different than "I'd like to leave the room, please"). She tried this on a third occasion while I was emptying dishes from our dishwasher. On that occasion, I simply told her that she had an alternative pathway to leave the room and that she could use that or wait until I was finished unloading the dishwasher. At that point, she stopped these accusations because she knew that I knew the law and I'd be avoiding that trap.
My ex-wife developed a legal theory that I was required to live in the school district of our children. Not near her home. Not within a state. But, within the school district of our kids. She has complained that I failed to listen to her on this issue at least 10 times in the 5+ years since we separated.
Also, be very prepared for her to get angry at you when you start to move on. It's amazing how jealous cheaters get. My ex-wife absolutely hates my current wife. She can barely speak to her and she only does so with a clenched jaw. I didn't meet my current wife until my ex and I had been separated for two months, at which time my ex was dating her now live-in boyfriend (I don't know why they aren't married) while continuing her exit affair (with a 21 y/o COW), having an affair with her friend's husband, and having a tryst with my kids' soccer coach (I had access to her cell phone records).
5. Get your finances in order.
I'd recommend that you split all liquid assets immediately (50%-50%) and keep your half in a place where she cannot get to it. I'd recommend that you also separate your credit cards and other joint assets. Be careful to make sure that she still has the ability to spend money and to live a normal life.
You do not want to appear punitive. You are not doing this to punish her. You are doing this to separate your lives. Do not cancel any insurance policies of any kind without her written permission.
Do a credit check. I got myself an account on credit karma (later than I should have) and I noticed that I was still on one of my ex's credit cards even after our divorce. This would have affected my ability to get a loan if I hadn't caught it in time.
Look at any retirement accounts or other investments that you have.
Learn about what is defined as a "marital asset" versus a "non-marital asset". In my State, a non-marital asset was any possession that I owned prior to the marriage (for example, my canoe!) or any gift that given my a third party specifically to one of us. So, her parents once gave me a DVD for Christmas. That was not an asset that was divided as part of the divorce because that was non-marital.
Combine all of these together. My retirement account (from my job) started about a year before she and I got married. Those additions to my retirement account and all of the interest earned by those deposits were non-marital assets. Unfortunately for me, I didn't keep those records and neither did the company that manages my retirement account.
6. BE THE BEST FATHER THAT YOU CAN BE.
This is THE MOST IMPORTANT PIECE OF ADVICE.
You are scared. Your kids are even more scared, I promise. Their entire world is mom and dad together, which ending. So, go out and do stuff with your kids. It doesn't have to be fancy or expensive. That is, you need to spend time with your children (you don't need to spend money on your children).