The easiest way to adopt in the US is to foster first.
This is a difficult area to say the least. The fact is that the way you feel about having more children can change rather dramatically. Those first months/year are not by any means easy. Pregnancy and delivery are not very easy. I think in part the reason for this is so a woman doesn't want to get pregnant right away!
I wouldn't do anything permanent to prevent pregnancy. I would talk to your wife's caregivers about why her pregnancy/delivery were so difficult to see if those factors are likely to occur again. Then I would use a temporary method to avoid pregnancy (i.e barrier methods etc) for a good year before doing anything long-term (i.e implant, hormones etc).
Part of the problem with hormonal and more permanent methods of birth control is that they have side effects that are undesirable/unhealthy. Our bodies were made by God to work in certain ways. The long-term implications of physically disrupting the passage of the egg to the uterus, and a vasectomy can be very, very bad. When you look at life long term you realize that sometimes the risk isn't worth the short term benefit. I have another 10 years at the MAX of fertility. I hope to be able to share intimacy with my husband a great deal longer than that. If I was told that another pregnancy/delivery would be dangerous my husband or I would do something permanent. But at this point the chance of another pregnancy to me is less dangerous than the side effects of a tubal ligation or vasectomy. The side effects of a vasectomy are greatly ignored. It is seen as an easy option because it is day surgery. But it can have lifelong debilitating side effects for many men (permanent swelling, pain and other effects). For that reason I don't think it is something that should be done unless there is a medical need.