Washington DC to Ljubljana

Fastest? Cheapest? Either might depend on the specific date you are flying.

I'd start by looking at flight options into Ljubljana on Google Flights. If they didn't look good, I'd also investigate flying into Zagreb, which has bus and rail service to Ljubljana, taking between 2 and 3 hours. Flying into Trieste might also work if you get lucky with flight connections. I think there is shuttle service available from Trieste to Ljubljana. I suspect Zagreb would be easier than Trieste as well as less expensive, because there's fairly frequent scheduled transportation.

Zagreb is a very nice city itself. I haven't been to Trieste, but I think it's worth visiting as well.

Probably on Air France or Turkish Airways out of Dulles, each of which appear to have connecting service into Ljubljana.

Frankfurt is quite close to Ljubljana, and there are plenty of reasonable flights from between the two cities. There are a million flights to Frankfurt from everywhere. Go nonstop to FRA and make one easy connection.

I flew to Venice, spent night, and then took early morning bus to Ljubljana. (About 3 1/2 hours, $30 on Flixbus)