JR Line (Ltd. Exp.): Shin-Osaka - Tennoji - JR Wakayama (about 60min.)
JR Line (Rapid Service): Tennoji (in Osaka C.) - JR Wakayama (about 70~80min.)
Nankai Line (the fastest): Namba (in Osaka C.) - Wakayama-shi. (about 60min.)
Both lines are connected with Kansai International Airport (KIX).
JR Kinokuni Line is convenient for tourists to go down to the southern part of Wakayama Prefecture including Kumano Region. JR has a limited express service to Shingu. The limited express trains are called " Kuroshio." The name means "the Black Stream (in the Pacific Ocean)." in Japanese.
JR Wakayama Line from Wakayama Station takes you to Hashimoto near Koyasan. If you change trains there, Nankai Koya Line takes you to Koyasan. (Many of visitors from Osaka to Koyasan use Nankai Koya Line from Namba.)
My Case (as of December, 2022)
As I live near JR Wakayama Station in Wakayama City, when I need to go to Osaka City, I usually use JR Rapid Service from JR Wakayama Station to Tennoji Station (80 min., 840 yen) or Osaka Station (88 min., 1270 yen).
I sometimes use Nankai Line from Wakayama-shi Station to Namba Station (59 min., 930 yen). Especially, when I feel like sitting in a relaxing seat all the way, I purchase a limited express ticket (520 yen) of " Southern (Sazan in Japanese)." "Wakayama-shi Station" means "Wakayama City Station" in Japanese.
From/To Kansai International Airport (KIX)
When I have a heavy piece of baggage, I use a airport limousine bus service from JR Wakayama Bus Stop (at the east entrance of the station) to the airport (40 min.,1200 yen). When I don't have to carry anything, I use JR Line to the airport (43 min., 900 yen). Of course, I can go to the airport by using Nankai Line from Wakayama-shi Station (47min., 880 yen).
From/To Shin Osaka Staion (the Shinkansen)
When I use the Shinkansen, or Japan's Bullet Train, I always use limited express trains named "Kuroshio (the Black Stream)" to Shin Osaka Station. The service is convenient for me because I don't have to change trains.