Yes, I meant Wakayama City. Or more specifically, Wakayama station, in Wakayama city. If you take the normal trains (which are only a little slower than the Limited Express options, even from Kyoto), then you need to change once, in Osaka Station. From Kyoto, you'll need to catch a Rapid or a Special Rapid. The Special Rapids normally run every 15 minutes, but there are fewer of them early in the morning. They take 30 minutes Kyoto-Osaka stations. The Rapids take just a little longer, some 36 minutes or so.
In the massive Osaka Station, you then need to head to Platform 1. Since it's the airport train, it's very well sign-posted. Trains run every 2-3 minutes at platform one, and only some of them go to Wakayama. But it's easy enough to figure out.
Shin-Osaka has Limited Express services to the airport, and also to Tanabe, but the regular airport/Wakayama trains don't stop there.







