Welcome to SI and sorry that you've had to join the best group that nobody wants to join. First, the resources blurb. There are some posts pinned to the top of the forum that we encourage new members to read. There are some non-pinned posts with bull's eye icons that are good resources, too. (I recommend the ones on consequences.) The Healing Library is at the top of the site and has a ton of great information.
If he wants R (reconciliation), then he needs to read How to Help Your Spouse Heal From Your Affair by Linda MacDonald. It's a nice starting point for him to follow. (I read it and agreed to most of the items.) Another good reference book is Not Just Friends by Dr. Shirley Glass. This one may be good for the two of you to read together and compare thoughts/feelings.
One of our members is really good at pointing out that it isn't always the A (affair) that ends the M (marriage), but it's the actions after the A that end the M. Basically, lying and TT (trickle truth) are the straws that break the camel's back.
Both of you should have STD/STI testing done. There are some nasty things out there, and they are health hazards for you. HPV can be brutal, as well as other diseases that are out there.
Ask him for a timeline, with details about thoughts & emotions around those things. Another thing may be to work on a full therapeutic disclosure that he does with his therapist. As for MC, please be careful as many use the "unmet needs" fallacy and may subtly shift the blame to you. The A was entirely 100% on him. Many MCs shift some of the blame to the BS (betrayed spouse - you) and that's not true. If we went on the unmet needs portion, I'd have been the one who cheated.
Cheating is more of a character flaw. If your WH (wayward husband) isn't willing to do the work, then it may be really hard to R (reconcile). Both partners need to be in 100% because it's hard work.
If you can, in an age-appropriate manner, let your kids know that there is an issue that you and your WH are working on and none of this is due to them. It isn't something they did or didn't do (that advice goes to you, too) and that there are some things that mom & dad are working through.
We call it the emotional rollercoaster and it can take you for a ride at any time. For me, the first year was horrible. I would go out to my car and scream & cry till I was hoarse. The second year, I'd still have ups & downs but could control the anger a little bit more. But - at about 18 months, XWH (wayward ex-husband) crossed my hard boundary and we were headed for D (divorce).
My second therapist was a betrayal trauma specialist and we worked through The Grief Recovery Handbook, which really helped me release a lot of my anger.