The information below concerning the international arrivals is the general procedure for most US airports. There may be slight modifications for Detroit, and I welcome corrections in the comments. I have flown through Detroit many times but I have never personally arrived there on an international flight.
All1 international passengers arriving in the United States pass through US Customs & Immigration controls after their first flight lands in the United States. So when your daughter gets off the plane in Detroit, she will be guided to US Customs & Immigration control. Note that if your daughter is not a US or Canadian citizen, she will either need a visa or an ESTA; if you do not already have plans to get her one, I would recommend asking about this as a separate question.
As part of the customs process, your daughter will need to re-claim any checked baggage. Once she has passed the customs checkpoint, there will be a special baggage check desk for connecting flights; she should re-deposit her checked bags there so that they can accompany her to her final destination.2 If she doesn't check any bags, she can skip this step.
After your daughter has re-checked her bags, she will exit into the "landside" (unsecured) part of the terminal. She will then need to go through a security checkpoint to get to her domestic flight. I believe the security checkpoints are up a few levels from where she exits the customs area.
Once your daughter is through the security checkpoint, she'll be back in Terminal A. She should admire the fancy fountain, check the screens to make sure her gate hasn't changed (it almost certainly will between now and the flight), and then proceed to her gate. Terminal A is very long — about 800 m between where she'll enter the terminal and the farthest gate — but there is an "Express Tram" that runs on an elevated track in Terminal A, which she should use if her gate is near the end of the terminal and she doesn't want to walk that whole distance. If her departure gate is in Terminals B or C, she will need to descend into a tunnel which runs under the tarmac (about 200 m) and then proceed to her gate (no more than another 300 m).
Finally, your daughter should keep in mind that the return connection will be a little different. There are no exit immigration controls in the US (as there are in Europe); Detroit does not have a separate international terminal; and all Delta flights use the same terminal. So (assuming your daughter is connecting through Detroit again) she will get off her flight from North Carolina, remain in the secure area, and proceed directly to the gate for her flight to Amsterdam. There should not be a need to claim baggage, go through a security checkpoint, or talk to any customs & border officials in Detroit on the return trip.
1 Except for certain flights arriving from Canada, the Caribbean, Ireland, and the UAE; but that's not applicable here.
2 This is the part of the process I'm least sure about. I have a vague recollection that some airports have systems where the customs officer shows you a picture of your bag on a screen but you don't actually physically claim it. I would recommend that you or your daughter ask the folks in Amsterdam whether she'll need to re-claim her bags to go through customs in Detroit; and if she's not sure what to do, look around for uniformed Delta staff and ask questions.