I visited Yoron Island (Yoron-jima) in Kagoshima Prefecture at the end of January, 2018. In this article, I will tell you how to get to Yoron Island from Osaka (as I live in Osaka) using airplane. Anyway, to get to Yoron Island, first you need to go to Okinawa or Kagoshima.
You can also visit Yoron Island using ferry, but it takes time and is not popular among tourists, so, it is not mentioned in this article.
1. Via Naha Airport (Okinawa)
I think this is the most popular way to get to Yoron Island if you visit there from Osaka. There are many airplanes flying to Naha Airport from Osaka including LCC (low-cost carrier), and I think the cheapest airline is Peach Aviation (Japanese LCC).

Airplane of Peach Aviation
If you use Peach, an airline ticket price from Kansai International Airport (Osaka) to Naha Airport is around several thousand yen, and they often have a sale. I booked the ticket for 2,990 yen on a Christmas sale.
You will arrive at a warehouse-like Naha Airport LCC Terminal building if you use Peach. Be careful that it takes time to transit from this LCC terminal. There is only one plane a day to Yoron Island (as of January, 2018), so you may need to stopover in Naha for a night. There are many reasonable hotels in Naha, so I recommend you sightsee Naha if you have the time.
From Naha Airport to Yoron Airport, JAL Group’s RAC (Ryukyu Air Commuter) operates an airplane a day (as of January, 2018). It takes approximately 40 minutes. The price of regular airplane tickets is over 10,000 yen, but if you make a booking earlier than 28 days prior to departure (SAKITOKU Advanced Purchase Fare), you can purchase a ticket for less than 10,000 yen (my ticket price was 9,550 yen).

Airplane of RAC (Ryukyu Air Commuter)
2. Via Kagoshima Airport
There are also many planes flying to Kagoshima Airport from Osaka, and I think the cheapest airline is Peach Aviation same as Naha Airport (around several thousand yen).
JAL Group’s JAC (Japan Air Commuter) operates a plane a day (as of January, 2018) to Yoron Airport from Kagoshima Airport. It takes approximately 80 minutes. The price of an airplane ticket from Kagoshima Airport is about double of an airplane ticket from Naha Airport, so I don’t recommend this route if you don’t have some plans in Kagoshima.
By the way, there is a plane flying from Amami Island (Amamio-shima) to Yoron Island via Okinoerabu Island (Okinoerabu-jima) in Kagoshima. I’m interested in this route as an island-hopping lover, so, I may take this route when visiting Yoron Island next time.
Conclusion
As a conclusion, I recommend you visit Yoron Island from Osaka using airplane via Naha Airport. And from Osaka to Naha, choose an airline or flight depending on your schedule or budget.