Zurich to Japan and South Korea for three weeks

Regarding the air-fare you have a few options. Buying only one-way tickets is generally the most expensive option, but it depends on the air-line. Since you are flying from Zurich, there are not too many low-cost air-lines available for inter-continental travel.

Option #1

@Tom already suggested buying a regional ticket (2 hour flight). I think this is your best option. You would fly return from Zurich either to Japan or South Korea and simply buy another return ticket from Seoul to Tokyo. To save time and flight distance, I would recommend to fly:

ZRH >> SEOUL >> TOKYO >> SEOUL >> ZRH

With 3 weeks of available time, I would spend about 6 days in Seoul then fly to Tokyo, spend about 12 days in Japan, return to Soul for another 2 days and then return to Zurich.

Option #2

You can also book stop-overs with some air-lines. This could even be cheaper than the previous suggestion IF you find a connection that stops in Seoul before going to Tokyo, which might be unlikely, but you can check online... Many airlines allow you to spend a few days (or weeks) as a stop-over at any destination along the route. This would mean you only buy one return ticket and arrange for a stop-over in Seoul:

ZRH >> TOKYO >> ZRH (via SEOUL)

With such arrangement you would be allowed / required to stay in Seoul for a few days instead of directly connecting to Tokyo.

Option #3

Multi-destination / international tickets are also available from some airlines, but they often include certain restrictions to the direction of travel (e.g. you can not have a return flight, but would go around the world through the US to come back to Europe). This only really makes sense if you want to see more places along the route and is usually more expensive than the previous options.

Well three weeks is not much time, you could easily spend that in either country and just scratch the surface.

For air travel, buying three one way tickets will likely cost a far bit more than buying a round trip ticket, plus a regional ticket. I would look at a couple of options: flying r/t ZRH-ICN or ZRH-NRT with a second ticket r/t between Japan and Korea; an open jaw ticket ZRH-ICN NRT-ZRH (or reverse) with a one way ticket to connect.

You could use the ferry in lieu of the regional Japan/Korea flight(s) but it will use up a lot more of your limited time, unless your intended places to visit include the seaport towns.

Where to go ... depends on your interests. Personally I like old Japan and Korea, getting out into the countryside or smaller historic cities where traditions still reign supreme. Love staying in Japan's traditional ryokans and minshukus. But the various neighborhoods in Tokyo and Seoul are also interesting to check out a bit.

I will add to downhand's answer, which I mostly endorse (especially the part about the ferry!) I will give some alternative suggestions about some of my favorite places off the beaten track, especially if you like walking.

I would personally spend somewhat less time in Tokyo and Seoul -- they are delightful places both, but in my opinion the tourist sights in both places can be missed. The fun part of these cities, in my view at least, is just walking around and seeing everyday urban life.

I'd probably skip Kobe, unless your itinerary is otherwise short on urban Japan. It is a delightful city (indeed, I am there now for work and having a great time!) but there are pretty much no tourist destinations.

In the outskirts of Nara, I would very strongly recommend walking all or part of the Yamanobe Road. I did all 31 km of it this past Saturday, and it's easy to do just half of that if you prefer (there are lots of train stations near the route). It walks through rural Japan, at the side of mountains, through small villages, past Buddhist temples and Shinto shrines, and past the ancient burial mounds of emperors from ages past. If you've ever seen My Neighbor Totoro, or any other animation by Hayao Miyazaki, imagine that it came to life and you got to spend all day walking through the set.

In Kyoto, there are lots of places off the beaten path. You should definitely do Shijo-Kawaramachi, Yasaka Jinja, Kiyomizu-dera, etc. like most other tourists, but my favorite place is Fushimi Inari Taisha. It is open at night, which is a wonderful time to go. A bit scary, in a good and exciting way! Also check out Arashiyama, especially if you don't otherwise see much of small town and rural-ish Japan. Nanzen-ji is another beautiful temple -- it supposedly closes at 5:00, but when I was there they only closed some hall with some Buddha statue or something and you were free to walk the breathtaking grounds in solitude.

If you like baseball, check out a game anywhere in either Japan or Korea. The most raucous sports crowds you'll see outside of South American football.

On your way west to Fukuoka, a convenient stop is Himeji -- which has the most beautiful castle in Japan (not a reconstruction like many others).

Once you're in Korea, you might visit Haeinsa Temple -- a beautiful; stay there overnight and visit first thing in the morning, before the crowds show up! In Gyeongju, rent a bicycle -- a fantastic way to see the outskirts of the city and the surpassingly beautiful countryside.

If you enjoy hiking, I recommend a trip to the summit of Mt. Seorak. It's a doozy, so if you want to day-hike it get started early (e.g. 6:00 am). I recommend starting from the north side and descending the south: at the south end of the climbing trail there is an oncheon -- hot springs -- complex and you can take a hot bath in a huge bathtub for two hours. After a long hike you will definitely appreciate this (not to mention a heaping plate of galbi). If you skip the mountain hike I recommend a trip to another hot spring bath at least once -- they are called oncheon in Korea and onsen in Japan.

You are in for a treat. Have fun!