Welcome to SI and so sorry you had to find us. There are some pinned posts at the top of the forum that have some good information, as well as some marked with bullsyes. The Healing Library has a lot of great info, and includes the list of acronyms we use.
Your WH (wayward husband) should pick up a copy of How to Help Your Spouse Heal From Your Affair by Linda MacDonald. It has a nice blueprint to help him know. Make sure to get the Linda MacDonald version. Another good book is Not Just Friends by Dr. Shirley Glass. It is a much longer book. I really like the chapter on Windows and Walls because it provides a good example of establishing transparency between you two (windows), and setting boundaries (walls) between those outside your M (marriage).
Please know that this isn't about something you did/didn't do, said/didn't say (fill in the blank). The decision to cheat was 100% his choice. Not a mistake. A mistake is forgetting to grab a gallon of milk at the store. An A (affair) is lots and lots of conscious decisions to betray.
Please get tested for STDs/STIs, and he does too. He needs to show you his test results. He probably didn't use protection, and there's all kinds of nasty things going around that can put your health at risk. If you have trouble sleeping or with anxiety/depression, ask for some meds. You may not need to be on them forever, but they can help you get through this phase. Also, we talk about the emotional roller coaster, so a lot of what you're feeling and going to feel is normal. Well, as normal as can be in this abnormal situation.
When you think about the kids, remember that the relationship you have is their role model. My kids were adults when I D, and 2 out of 3 said I should have done it long before. They know when things aren't right. It's tough either way.
I understand his job is important, but I can't imagine choosing anything over our family
Yeah, I don't understand it, either. But the thought processes of a WS (wayward spouse) can be disordered, meaning they don't think the way a normal, moral person would. But I should be careful: with the amount of infidelity in the world, maybe it's my thought process that's disordered in thinking that when you exchange marriage vows to forsake all others, you're making a vow that you plan to keep.
If you can, IC (individual counseling) with a betrayal trauma specialist can be helpful. Generally we say IC for each. You to heal from the trauma, the WS to find their whys and work to be a safe partner. Then, MC to work on the relationship. I'll add my two cents. I'd done IC and XWH did IC (I think. Cheaters lie a lot.) We did some MC and one of the first things out of the MC's mouth was, "We'll look at why Mr. Fields decided to go outside of the M to get his needs met." I told the MC that I respectfully disagreed. If we were going to look at the unmet needs fallacy, then I should have been the one who cheated. (You can click on my person icon to read my story.)
Take time to recover. Take the time to take care of yourself, and to be there for your children. R (reconciliation) is a lot of hard work and many WS don't have the fortitude to do the work. Some people stay together just for the children, for their lifestyle, and many other reasons. The decision to R or D is up to you, and we'll collectively help you with advice. Our goal is to help you get out of infidelity.
Please keep posting. Keep the same thread if you can, as that helps us catch up on your story. Another saying we have is to take what you need and leave the rest.