Depends what you wanna see and do. A week is not much. The best option, I think is to start in the north, with Belgrade and Novi Sad. Devote 2-3 days to that. Then go south towards Macedonia. I've been to Macedonia when I was a kid so my memory may be fuzzy but there's not much to see there. City of Skoplje and lake Ohrid. Let's say one day in Macedonia and from there you can go to Montenegro. They don't share a border, you'll have to go through Serbia again (or Albania) but it's close. In Montenegro, don't spend too much time in Podgorica, there's really not much to see or do there. Just pass through, make maybe a couple of hours break. Go to the seaside (half an hour drive from Podgorica) and work your way up - cities of Budva, Kotor and Herceg-Novi. Herceg-Novi is north and very close to Croatian border and when you're there it would be a shame not to go to Croatia and see Dubrovnik, it's an hour drive from Herceg Novi.
That would make your trip look something like this:
It may be better to skip Macedonia entirely and spend more time in Serbia, Montenegro and Croatia, but that's up to you. It's a rather long detour. On the other hand, since you're American, you could go from Macedonia to Montenegro through Kosovo, just wear a piece of clothes with American flag on it, or you could go through Albania.
Depending on your funds, you could fly from Belgrade to Skopje and Skopje to Podgorica, that would save you a lot of travel time.
I don't really know what you like - big cities, nightlife, nature, rural areas, medieval cities, Roman stuff, ancient cities, renaissance cities...
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