Kyushu itineary

Planning to visit Tokyo + Kyushu last 2 weeks in December. Need some help with itineary on Kyushu.

We want to hit the main tourist spots + do 1-2 hotspring trips - how many days are enough? And what's the best way to commute?

Specific questions:
1) Is one day for Fukuoka enough, if we want to see Hakata, Tenjin, and adjacent areas? Are these close to each other and can be doable in 1 day (is it bigger/smaller than say Shibuya or Shinjuku for my reference?)
2) Is one day for Nagasaki enough?
3) Is Huis Ten Bosch worth it? We just want to go there and take some photos of the lights, but not sure if it's worth the admission
4) Is Kumamoto (Castle) worth a visit (in a day) if the castle is closed due to earthquake? We would go if we can still see the outside of the castle for picture
5) Which cities should we rent a car and which cities can we get by with trains?
6) Planning to go to Beppu for hotspring. What's the best way to get there?

You are not giving a lot of details so some things are still unclear. You are only staying in northern Kyushu so the northern Kyushu Rail Pass for 5 days might work out well for you.
https://www.japan-guide.com/e/e2361_05.html
All the places you mention are easily accessed by train - but if you go somewhere like Kurokawa, you'd need to take a bus or car. And you don't say how many "we" are - a rental car might be an economical and convenient way around if you are up to driving in Japan. There is also the Sun-Q bus pass to consider.
For Fukuoka, you can see most of the best in a full day. Do not listen to anyone who says there is nothing there - it's no Kyoto of course, but they just don't know what is there. The Nanzoin Temple is one of Japan's finest - yet nearly unknown to all the tourist guidebook "experts". It's also free and just 20 minutes by train from Hakata Station.
http://www.thejapanfaq.com/fukpic.html
http://bit.ly/1KAm8cU
Ohori Park is quite nice, and the traditional Japanese garden that almost nobody knows about there is really beautiful and relaxing. Nokonoshima has a nice island park and good hiking, plus there are many traditional villas and gardens like Yusentei, Rakusuien, Shofu-en, etc. The ramen at a stall (yatai) in Nakasu by the river is quite an experience as well.

One day for most of the best of Nagasaki is possible, but you'd need an early morning start and it's 2 hours by train one way. It would be far easier and better if you did an overnight there.
http://bit.ly/2ob4u8i
Besides, if the weather is good, one of the highlights is seeing the city from atop Mt Inasa at night for the "$10 million dollar view". It's regarded as one of Japan's best. And if you're going to see Huis Ten Bosch, staying overnight locally makes even more sense.
Opinions vary about Huis Ten Bosch - some love it, others find it a big Potemkin Village. For myself it was nice but what really makes it worth it is when visiting during one of their flower festivals - particularly tulips, hydrangea, etc.
https://youtu.be/JIKAjc5MfPs

For Kumamoto Castle of course you can see what you can from outside. Only you can decide if that's worth it. If you are going to Japan again someday in the near future though, I'd skip it. If not, you can go see it; the Suizenji Garden is the other gem there as well.
https://youtu.be/K9xyXmU8U1g
As far as getting to Beppu goes, you don't say exactly from where or how. If by train from Kumamoto City, you can take the bullet train only up to Hakata Stn and then take the Sonic Ltd Express - including the wait times it'd take about 2½-3 hours. Bus or driving might be better.

There are lots of guides and forum threads on Kyushu. Kyushu is a big place, and the first thing to decide is how much of it do you want to see, and how many days do you have?
https://www.japan-guide.com/list/e1108.html
https://www.japan-guide.com/forum/quereadisplay.html?0+164399

To answer your specific questions:
(1) Fukuoka (Hakata) area offers a lot to see. What's your interest? I stayed 3 days just to try different food Fukuoka is famous for. I ate at a yatai, a restaurant that specializes in serving squid sashimi "live", mentaiko, tonkotsu ramen, etc. I would not leave Fukuoka without seeing Dazaifu Tenmangu Shrine.

(2) 1 day for Nagasaki was enough for me. Many Japanese find Nagasaki exotic for its history and foreign influences, but the same things for tourists from foreign countries are not that exotic.

(4) Kumamoto castle, after the earth quake, is probably not worth seeing. When I went, the area was fenced off and none of the repair work had started yet, and it was a sad sight. But I was glad to have helped the area a little by staying there and donating to their economy. I enjoyed riding the steam locomotive to Hitoyoshi.

(5) Most areas (certainly the tourist spots) are accessible by train or bus. Some of them don't run frequently. Bus as long as you know their schedule, it's not an issue. I did not need a car, but if you are in a hurry then a car comes in handy.

(6) You can access Beppu by train or bus. It's an easy train ride from Hakata (Fukuoka). Besides enjoying various onsen, you'd want to tour the "7 hells".

Thank you Ken and psi5! Based on your feedback, I am now thinking of the following itinerary. Will appreciate your thoughts:

Background: 4 of us; 2 adults 2 kids (age 6/9). Been to Japan many times but never in Kyushu. Will stay in Japan for 14 days Dec 22 - January 5 (14 days).

Day 1 in Kyushu: Fly in to Nagasaki from Tokyo and stay in Nagasaki
D2) Take early train to Fukuoka and spend 1 (or more?) day there; stay in Fukuoka
D3) Train to Kumamoto and stay there
D4) Rent car from Kumamoto and drive to Aso in the morning, then drive to Kurokawa afternoon and stay there
D5) Drive to Beppu and stay there
D6) Drive to Oita airport in the morning, return car, and fly back to Tokyo

Questions:
1) I plan to take the Yokoso domestic flights so I don't need to travel back to Fukouka or Nagasaki after Beppu. Based on my route can you please confirm I am better off flying into Nagasaki and flying out of Oita?
2) Nanzoin in Fukuoka looks cool! How long should I allocate if we want to go there? Can I still fit it plus rest of Fukuoka in a day? Or does it mean I should extend my Fukuoka stay for 2 days (which is fine)?
3) For Huis Ten Bosch around December, are there any flowers left to see? Or it's mostly night view (LED lights)? Which is more recommended if we only want to stay 1 day in Nagasaki? HTB or Mount Inasa for night view? Or can we do both in same night?
4) For Kumamoto, besides the Castle, is there enough to do to spend 1 day and stay there?
5) For my day 4-6 in Kyushu, if I want to see Aso/Kurokawa/Beppu, am I better off with car rental? I have driven in Hokkaido before during the summer, but unsure about the road conditions in Kyushu and during December (will it snow/rain?)

Thanks again!

JAL Yokoso airfares have a black out for when you are in Japan - you can't use it. You can however use ANA's Experience Japan Fare.
http://www.jal.co.jp/yokosojapan/
http://ana.ms/20Xtf8t
It is the same fare with no black out dates - but what you should know is that there are no changes, re-routes or refunds with the fare. It's the same fare regardless of your routes, so flying into Nagasaki or Fukuoka or Oita makes no price difference.
For Nanzoin, it depends on your pace, and I am not sure how fast or slow your kids will be for example. But 75-90 minutes I think would do it. You can still fit in a lot of other places in Fukuoka City for the same day. I have not been to Huis Ten Bosch in December. It won't be as good as springtime of course, but I think they try to make up for it with all the lights. You can see both it and Inasa - you'll be in Japan during the shortest days of the year, and you can drive up to the top of Inasa anytime you want. Though if it is like last year, HTB might be something for New Years Eve.
https://is.gd/ePfOUl
For #4, I really don't think so. There are other options if you are driving though - there is the Kyusendo Limestone Cave, the biggest in Kyushu, and maybe your kids would especially like it.
https://is.gd/FlU8ZL
Going around Aso by car is absolutely ideal. It's still iffy if the crater area will be open, so check the following before heading out.
http://www.aso.ne.jp/~volcano/info/
https://youtu.be/Tyh_nKrv5tY
There are many additional sights that only a car will take you to - the Nabegadaki Waterfall is wonderful, although there are no clear signs for the side road to take you there (better practice virtually driving down the road with Google Maps first) and the Daikanbo Lookout is just spectacular.
https://is.gd/ZV8940
https://is.gd/vbYsK8
There are a couple other issues - some can rental places have some stiff drop off charges if you return the car to a different location. You might want to check on that. Also, around the end/start of the year, many places are closed. Finding a restaurant might be very hard on New Year's Day. The convenience stores like 7-11 can save your life. Always check with the website of a place to make sure it will be open around that time.

Hi. If you dont have a rental car, I think 1 day in Fukuoka is heaps. A day in Nagasaki city is plenty too although there are interesting side trips. Nagasaki is amazingly hilly, a bit like Hong Kong island. I'd spend the night there.

Huis Ten Bosch is not worth the trip unless you are using it as a base to go to the Nagasaki Bio Park (the most awesome place in the universe), or have a passion for dutch stuff.

I would try to factor in Yufuin which is a lovely little town and relatively easy to get to from Fukuoka or Beppu.

That new itinerary is way too rushed. You are really only giving yourself 4-6 hours at each location. The rest of the day is moving using long distance transport.

Thanks again EVERYONE! Based on the feedback I have adjusted the itinerary. Please let me know if I miss anything and some new questions:

Day 1 in Kyushu: Fly in to Nagasaki from Tokyo early morning and stay in Nagasaki. Tour around the city + Mount Inasa (Huis Ten Bosch and Bio park seems too far from the station? Guess will reserve it for next time :/)

D2-3) Take early train to Fukuoka and spend 2 days/2 nights there - food/shopping/sight see around Hakata/Nanzoin/Tenjin/etc.

D4) Train to Kumamoto and stay overnight

D5) Rent car from Kumamoto and drive to Aso in the morning, drop by Nebegadaki Water fall, then drive to Kurokawa late afternoon and stay in onsen

D6) Drive to Yufui in the morning, spend a few hours (?) there, then to Beppu early afternoon, tour the 7 hells, and stay overnight

D7) Drive to Oita airport in the morning, return car, and fly back to Tokyo

New questions:
1) How's the airport service during 12/31, if we fly back to Tokyo on that day? Or should we shift it to 12/30? One year we were in Tokyo on 12/31 and most of the shops were closed after 5pm, but if we have no problem flying in and accessing things like limobus, then it could be a good option for this "quiet" day

2) For my 2 days in Fukuoka, where should I stay that's centrally located to everything? We mostly enjoy the food and shopping in Fukuoka. Is Hakata a good location?

3) Does it make sense to get the Kyushu Rail 3-day pass for days 2-4?

4) For the car drive D5-7, how easy is the drive compared to Hokkaido which I experienced? Are the signs clear and straightforward if I have the rental GPS?

5) How many hours to plan for Nebaegadaki Fall and Yufui? Do you think that's doable per my itinerary?

You've made it no better. You still have a rushed itinerary. You to 4 hours of travel per day. Leaving you only 4-6 hours for sightseing.

Use google maps to calculate car travel times etc.

Nagasaki airport is 45 minutes away from Nagasaki itself.

I think you don't leave enough time to the onsen experience wanting to visit 3 onsen places in less than 48 hours. My suggestion would be to stay in Kurokawa 2 nights, or at least to spend a full day in Kurokawa touring different onsen and then drive out in the evening, if you absolutely must leave Kurokawa and move on. I would also select either Yufuin or Beppu so you can explore one of those two place in more depth.

Regarding driving in Kyushu, I found it quite normal, but I think Hokkaido is specially relaxed with those big straight streets. Kyushu is a bit different, but definitely no problem to drive there.

I feel bad adding something to your itinerary, but you might consider a day trip from Fukuoka to Dazaifu, if you like temples with zen gardens, or shrines or important cultural museums.

Enjoy your trip to Kyushu.

Maybe most easy is to remove Nagasaki (instead fly to Fukuoka) and use the free time for the other things.

Thanks again everyone for the feedback. Actually for my D1-3 in Nagasaki/Fukuoka/Kumamoto does it make sense for me to drive a rental as well? Or are those cities more populated like Tokyo so I am better off taking trains?