Any way you cut it, this is going to be a long trip. I would guess that your best connections will be at the following airports:
- NRT (with a bus trip to HND for the subsequent domestic flight)
- KIX (potentially with a bus trip to ITM for the subsequent domestic flight)
- NGO
- ICN (with a likely overnight stop at ICN)

I would probably choose something like this:

--- Pittsburgh, PA to Nagoya, Japan (20 hrs 0 min, 7561 miles)

United Airlines Flight UA5814 on a Canadair RJ 700 (jet) (operated by UNITED EXPRESS/SKYWEST)
Departs: Pittsburgh, PA Sat, May 12, 6:00a (1 hr 35 min)
(PIT: Pittsburgh Int'l)
Arrives: Chicago, IL Sat, May 12, 6:35a
(ORD: Chicago O'Hare)

--- Layover in Chicago: 1 hr 25 min ---

United Airlines Flight UA831 on a Boeing 777 (jet)
Departs: Chicago, IL Sat, May 12, 8:00a (4 hrs 24 min)
(ORD: Chicago O'Hare)
Arrives: San Francisco, CA Sat, May 12, 10:24a
(SFO: San Francisco Int'l)

--- Layover in San Francisco: 1 hr 16 min ---

United Airlines Flight UA831 on a Boeing 777 (jet) (lunch, movie)
Departs: San Francisco, CA Sat, May 12, 11:40a (11 hrs 20 min)
(SFO: San Francisco Int'l)
Arrives: Nagoya, Japan Sun, May 13, 3:00p
(NGO: Nagoya Chubu Centrair Int'l)

Note: Although this is published as a "direct" flight, you must change planes at the connection in San Francisco Int'l (SFO). Note: You must change terminals in San Francisco (SFO).

And then ....

All Nippon Airlines
ANA 1827
Departs NGO 16:20
Arrives Matsuyama 17:25

For the return, you can find a return via NGO. There's an ANA 1822 flight from Matsuyama. It reaches NGO at 14:00 - one hour before the UA departure to SFO. That connection might be a little bit tight - but I think it's doable. Better check with a travel agent. (And better buy tickets soon)


Dogo onsen.
Nuclear power plant in Ikata.
Windmills (the modern variety)
Hire a car and drive around...