Traveling from US to Slovenia cheapest way

I don't live in Atlanta, but I know it's a Delta hub. So you should get plenty of direct flights to major European capitals (not Ljubljana though obviously).

I would check Google Flights and Skyscanner, starting with Ljubljana as your endpoint, and see what you come up that way first.

On past trips to Slovenia, I've had good luck flying into Venice and taking a shuttle from the airport directly to Ljubljana (DRD or GoOpti). That would make sense if you can get a direct flight to Venice but it might not be the cheapest. You probably won't get a direct flight to Zagreb, Croatia, but if you can fly in there cheaply enough, you can get to Ljubljana by train or bus in a few hours.

SkySkanner should give you options to stage in London or Paris etc. and connect to different flights like easyJet or Adria.

If price is most important, try flying to London (watch the airport changes), Paris, Amsterdam, Madrid, Frankfurt...and if you get a really cheap fare look for connecting flights on to Ljubljana on your own. (Maybe a train or bus to Ljubljana if you can get a super cheap flight to say Vienna or Munich.) Often it will be easier in that case to spend a night in that city going home before flying back to the US - still might be cheaper than trying to book one ticket Atlanta to Ljubljana. All depends on time of year, origin city, etc. Atlanta may be a very different fare structure than me flying there from Portland, so what worked for me may not work well for you!

Turkish is the cheapest mainline carrier with a nonstop to Istanbul and connecting flight to Ljubljana. Some people (I have not flown them myself) love Turkish for the food and service.

Any of the big on-line air agencies will do a better job of assembling itineraries than you can on your own. You can start with matrix.itasoftware.com (but no tickets sold) and pick several others for comparison. They lay out the possible connections, time in the air and on the ground, and of course the most important, price.