In providing commercial janitorial service, we're attuned to personnel screening and security, and (given our Arizona location) we deal with many immigrants. So I can't resist the urge to comment on the current European refugee crisis.
Second, communities facing genocide ought be at the top of our list. Those facing economic hardship, or political repression, have serious problems, but not at the same level. Examples of genocide would be various denominations of Christians in Syria and environs, and Iraqi Yazidis (a Kurdish religious community of Zoroastrianism and ancient Mesopotamian origins).
Third, within a population, we ought prioritize the most vulnerable, both those generally vulnerable and dealing with the proclivities of many near eastern polities - women and children. We also ought keep family groups together, both for obvious reasons and for the good of the children.
Fourth, screen as possible. Many will be without documents, and what documents are available will likely be suspect. Intensive interviewing of family members, separately, ought help establish both family legitimacy and circumstances of persecution.
Fifth, avoid unaccompanied adult or late teenage males. As the least vulnerable of the refugees, they ought be far down our list anyway, and they are most likely to be current or potential terrorists.
Family groups, kept together, are most likely to be hard working, productive residents (parents trying to provide a life for their kids), and least likely to be societal problems for us, both short and long term. Finally, I suspect minority groups that have survived for millennia by adapting to dominant regional cultures might be most likely to adapt to American culture as well.
Regarding worries that ISIS (or another group) might insert a terrorist into the mix of those I suggest we admit, I'd note that there are easier ways to get into America (a tourist or student visa, for instance), so we would not exposing ourselves to much more danger than currently. And, again, the profile of those I suggest we admit does not lend itself to eventual radicalization, while we do seem to produce quite adequate numbers of radicals in our prisons and mosques and on the internet.
Finally, protecting the persecuted is the right thing to do.
PS - Here's an article from the journal First Things on the Vietnamese refugee resettlement operation back in the 70's. Note the numbers involved, the settling of entire family groups, and the speed at which it was accomplished.