Be prepared: You will uncover lies, and no one will believe you. (You will probably not believe it either. It may take years for you to accept your reality.) Your husband is among the world's biggest liars (only a slight exaggeration), and he lies without any sense of right or wrong. In fact, he probably believes the lies he manufactures.
Begin to document everything. If you get a divorce, you will need this to protect yourself because he will lie about YOU. Even if you think he would never do that, you need to prepare as if he will.
Keep copies of all emails and text messages he sends you. Delete nothing.
If possible, do a complete download / backup of the computer and the cell phone he has been using. You may be able to get download copies of things like Facebook Messenger, Apple Messages, or other messaging. Look for pictures, video, emails, anything that gives you a trail of his deceit.
Do not wipe the hard drive clean! (I did, stupidly!) Make sure you save printouts or digital copies of everything you find.
If you can handle it—or if you need to see it to believe it—create a new email addres and sign up on Match.com, Tinder, AdultFriendFinder.com or whatever is the current hook-up site of the day. These are easy ways to find your cheating husband online. Forget about monitoring him; this is documenting his activity so that when he denies it later, you have the proof. Caveat: the danger is that now you have an account ... ick.
If you can afford it, hire a private investigator who specializes in digital forensics to do the research for you. That will only hold up in court if the PI testifies in person.
Some of this initial gathering of proof of his activities is for you to see for yourself the depth of his deceit and addiction. Your first inclination may be sympathetic and forgiving. Once you have seen him online, you may feel less sympathetic. That's a good thing. It can be a jarring wake-up call.
Any documentation you collect, I urge you to save. I repeat: do not be tempted to get rid of it. He (or The Experts) may indicate you should move forward, that hanging on to this evidence shows that YOU have a problem with forgiveness. IGNORE that bad advice! The evidence you collect may be very important in the future if HE tries to build a case against YOU.
Update 15 years later: You may find that all the evidence of sexual misconduct is meaningless in later custody litigation. That doesn't mean you shouldn't collect it; Cover all your bases by collecting documentation.