I am a lawyer, but not your lawyer, obviously.

I am a family law attorney. In my state, the law is very favorable to someone in your position. It may be in your state as well, assuming you're in the US. If at all possible, its probably a good idea to have enough funds to get through several months of attorneys fees. Even if it means taking it from a joint account (ask your lawyer for specifics in your jurisdiction). Many of my clients do not anticipate how much a divorce costs to get started, and they are unpleasantly surprised when they realize court costs and attorneys' fees for the initial paperwork sometimes use up your entire $2000-4000 initial payment. Custody is messy, and if your husband fights you, it could get quite expensive.

If I were in your position, I would try my best not to say anything until I was able to get enough money to live, care for my children, and pay my lawyer for at least 3 months. I would also disclose to my attorney exactly what types of proof I have of the infidelity. The way information/documents are collected matters in court; your attorney can guide you on the best ways to collect evidence.

In my state, you would receive spousal support for at least a year, but it would still take a few months to receive you first payment. Same for child support; it takes a while to kick in. If your state has alimony, make sure you can take care of yourself while you wait -- this is THE biggest problem in 75% of my client's lives, and my rates are looowwwwww.