Speaking as someone who lived through this, and after first separating and filing for divorce is now with DH in a stronger, happier marriage my advice is:
Separate for a while. You need space to grieve and heal, the hurt runs deeper than you can initially imagine, and he needs to know you are the one in control here. He doesn't get to choose who he wants to be with. So many men who have affairs have this warped notion they can string two women along while they decide who they want to be with. Errr no.
Tell family and friends. You will be amazed by the level of support they will provide, I was. It isn't your job to hide his grubby secrets. If he is embarrassed and humiliated by this, well he should be.
Speak to a lawyer and understand your legal rights. It is all part of taking back control of your life, and is very reassuring. Where you go from there is entirely up to you.
Think about your marriage, honestly. Is it worth trying to save it? Before you knew about this were you happy, or were you just rubbing along for the sake of the dcs? Getting back after an affair is bloody hard. I'll not sugar the pill. Betrayal is a very difficult thing to get past, even if your DH behaves impeccably going forward.
What needs to change? What do you need him to do? He needs to fall over himself to do whatever it takes. If he won't, you won't be happy, and you won't be able to trust him.
Don't fgs spend hours analysing what 'drove him to this', what you could have done differently, or anything along those lines. NOTHING you did or said made him cheat. NOTHING. Repeat that after me : NOTHING. He chose to because he was selfish, weak, self centred, and arrogant. If he was unhappy in the marriage he should have discussed it like an adult. I am sure you have been unhappy at times, and you stayed faithful.
Post here, you will get a lot of support. Not everyone is of the LTB mentality, many of us have been there before you, others have a lot of experience in this area and can give sage advice.