Kyushu area 2 questions

Kyushu- visiting Fukuoka 3 nights 1day city explore
second day visit either Dazaifu or Beppu hot springs? ?Help please Not sure which will be the most interesting.Then moving on to Nagasaki 3 Nights lots of sights in city and Gunkanjima island. Then Matsue 3 nights plenty to see here yuushien garden and vogel park and Adachi museum of art. Returning to Fukuoka final day for return flight.
Cheapest way of travelling ? would a pass be good value or pay as I go?
My main interests are cultural -gardens/temples shrines art galleries,nature.

second day visit either Dazaifu or Beppu hot springs? ?Help please Not sure which will be the most interesting.

It depends on personal interest. I personally prefer love hot spring bathing. Therefore, I would chose Beppu over Dazaifu by a wide margin. But a history freak who does not like to be naked in front of others will probably prefer Dazaifu. Another factor could be money. The side trip to Beppu is much more expensive than to Dazaifu.

Cheapest way of travelling ?

The cheapest way would be by highway buses. There is an overnight bus between Matsue and Fukuoka:
http://www.ichibata.co.jp/bus/highway/fukuoka.html

Fukuoka is a nice city to spend a couple days exploring. It is also not overrun with tourists either, and you can get to most of the best sites by subway. A good site with info on what to see is at:

http://www.tinyurl.com/see-fukuoka

Also you can either get a subway day pass or if you want to take buses also (as well as use it all the way down to Dazaifu), there is a tourist pass you can take advantage of til at least this September. See:
http://yokanavi.com/access_tourist_eg

Daizaifu is famous for its Tenmangu shrine but there is not else of real interest there. You could stop by and be through it within 30 minutes on your way if you want to, otherwise skip it and go straight to Beppu, which has far more places of interest.
Beppu is most well known for its hot springs of course but has many other good places to see, including its Hells Tour, Monkey Park/Aquarium, and view from Mt. Tsurumi (if the weather is good). There are some videos
at:
http://www.youtube.com/watch?v=K2NNi8MLcXM
http://www.youtube.com/watch?v=zepPfkuqhok
http://www.youtube.com/watch?v=cBnZneDsKqQ
and lots of useful info at
http://www.city.beppu.oita.jp/02kankou/english/01info/02shisetsu/index...
http://www.beppu-jigoku.com/images/English.pdf
You might want to look into a JR Kyushu Rail Pass - see http://www.jrkyushu.co.jp/english/railpass.html
It is very easy to recoup the costs of the Pass for northern Kyushu.
Nagasaki is really a jewel of a city and really the most, if not one of the most, interesting cities in Kyushu, with a lot of history and diversity. Gunkanjima is really unique but please know it will take up at least 3 hours of your day, and the boats may not go at all if the sea is too rough. Mt. Inasa is also spectacular, especially at night. Please see:
http://www.ngs-kenkanren.com/eng/
http://www.at-nagasaki.jp/yakei/en/index.php
http://www.at-nagasaki.jp/foreign/english/

The northern Kyushu JR rail pass is a bargain. It will get you to Nagasaki, Beppu, and Kumamoto.

I have just come from Takachiho, and had a great time. It requires bus travel to get there.

http://www.japan-guide.com/e/e8050.html

It is golden week, and I am not sure what parts of this are specific to golden week, but I will tell you about my day.

I stored my luggage in one of the bus stations lockers, and went on foot.

I visited Takachiho shrine, and it was very well kept up, with what looked like lots of very old cedar trees around it. There was a shuttle bus to the gourge for 100 yen, and I went to the main gourge area.

It was very festive, many people were renting rowboats into the gourge, but I stayed on the walkways above and took pictures. There were two stocked fishing ponds for people to catch fish, and the fish were gutted and cooked for you by the vendor. There was also a noodle shop operating "noodle streams", called flowing Somen. It was interesting to watch.

http://www.toyohara.com/inaka/somen-e.htm

After I took the shuttle bus back to shrine, I walked back to the bus terminal to go to the cave shrine where the sun goddess Amaterasu was supposed to hid herself away. I stopped in the information center accross the street from the bus center to find that they were teached a craft. They were hand carving Magnatamas out of pieces of marble. The kit was 1000 yen, and had everything needed except metal files, which the information center was supplying. I bought a kit, to take home with me, and then grabbed my luggage and took the bus to the Amano Iwato Shrine.

I dropped my bags at the Hana Ryokan (near the shrine), and visited the shrine. After going to the shrine, I went back to my Ryokan, and found out about a free shuttle that eveing to the Yokagura Dances at the Takachiho shrine.

The Ryokan had a 14 course meal. Well fed, I went to the dance all at the shrine.

The Dance hall fee was 700 yen, and the hall showed three of the standard Yokagura dances, and an additional one (a Goshintai Dance) about two gods making Saki and getting plastered. The last dance was the most fun to watch.

The Ryokan supplied free tickets to the Amano Iwato Spa, but I ended up finishing the day at their bath before going to bed.

In all, it was a very full day.

The meal in the morning was just as elaborate, and the people at the Ryokan drove me to the bus station after the breakfast to catch my bus.

If I ever come back to Japan during holy week, I plan to go to Takachiho again.

I am not sure how much of this appeals to you, but I thought it wouln't to mention this as an option.