US citizen go to Japan and Korea

If I'm correct in interpreting this as two separately-booked round trips (LAX-Osaka round trip with a separate Osaka-Korea round trip), then this might be difficult to do successfully due to the (short) 1 hour time between flights.

Getting to your next flight: You will generally need to pass through security when transferring. This takes a bit of time, but not too long. Additionally, for a separate booking, you will need to check in separately for your Osaka -> Korea flight. You might not have sufficient time to check in at the airport, with a 1-hour layover; it depends on the airline. You should definitely contact the airline flying you to Korea and ask if this will work.

Transiting through Japan: Since it sounds like you have separate bookings, sterile transit might not apply, meaning you might have to go through Japanese immigration before transferring to your Korean flight (even though you're not staying until later!). This will be somewhat tricky on a 1-hour layover. If you can't change the booking to allow for some more time between flights, then I would advise finding out whether there are any special procedures that would let you skip immigration during transit "on a separate booking" (i.e., specifically look into the "separate booking" phrase). I am not sure if such procedures are available in Japan or not.

Getting your bags to your next flight: If you have a separate booking, you will likely need to claim your baggage and then check it in with the next flight. Baggage claim has been one of the most time-consuming parts of the post-flight procedure, in my experience, and I do not expect this to be possible within 1 hour. If you do have a separate booking with only 1 hour between flights, I would advise flying with carry-on luggage only, and not bringing any checked luggage.

Procedure when entering Japan and South Korea (regardless of timing): both Japan and South Korea allow US citizens to stay as a tourist for up to 90 days visa-free (you did mean South Korea, yes?). Your total trip is much less than that, so this obviously applies. Therefore, no visa is required. However, from personal experience, Japanese Immigration staff will ask where you're staying, and Korea will likely do the same; if you've booked a hotel or have other accommodations, good! If not, it would be a good idea to arrange for accommodations before you arrive, at least for the first day or two. (I nearly made this mistake when I decided to "wing it" on my first time really being a tourist, but luckily I booked a hotel at the last minute)

Lastly, have fun with your trip! Hopefully you can arrange things to work. At the very least, call your airline, to see if they can assist.

Do I need to pass through security or do I just change airlines and go?

Security? Yes. Immigration? No. In KIX you'll pass through transfer security check (X-ray, metal detector, etc), and go onwards to your next flight.

On your way back from Korea, when you stay in Japan for two weeks, you'll have to pass immigration and customs, of course.