My oldest is adopted. Like dawgstudent, we went through IUI, IVF, ICSI, etc. in my first marriage and it's not something I would wish on anybody. That said, I wouldn't trade my oldest for anything.
Few things to know:
1. Each state is different and generally you're bound by the laws of the state where the adoption took place. I can tell you how things went for me in Texas, but wherever you live is going to be potentially different.
2. Unless you're going through CPS, the major difference between domestic and overseas adoption is that domestically the mother chooses you. That worked out well for me because we discovered the infertility at a pretty young age and most potential parents trying to adopt are older, so we stood out and matched very quickly. You essentially make a scrapbook with pictures of you and your wife, your interests, do you travel, if you have nieces or nephews, etc. and the birth mother looks through them and may want to meet you and your wife for an interview, then ultimately chooses the person/people they want.
3. Most adoption agencies that I talked to had basic guidelines for contact after the adoption (letters/pictures for 5 years, no direct contact, no last names, etc.), but really it's an agreement between you and the birth mother. As horshack mentioned, know what you're comfortable with in regards to post adoption contact and know what you can walk away from. We were pretty lucky and it was just the basic letters and pictures for a few years.
4. In Texas, with a domestic adoption, the birth mother can't sign the papers until 48 hours after giving birth. During those 48 hours, they may not even let you see the baby or they may want you hanging out at the hospital all day. Ours wanted us there when she was born, but I've heard a lot of stories that go a lot of different ways. Just know that is probably the most nerve racking 48 hours of the entire deal because there's nothing legally binding the mother to following through with the adoption. They can walk away at any point. Depending on the agency, not only do you lose out on the baby, you're also out some portion of your money as well.
5. With overseas, it's more of a queue. As long as you fulfill the country's age, health, marital, etc. requirements, you're in line. Depending on the country, that could be a few months to a few years. The biggest risk with overseas adoption are things like immediate changes in that country's adoption laws (this happened in Russia), civil unrest, travel restrictions, etc.
6. If you want an infant, as in a days old infant, you're going to want to go with a domestic adoption. If you go overseas, you're looking at a few months old to possibly a year.
7. It's really freaking expensive no matter what path you choose.
Feel free to DM and good luck!