nagasaki to beppu

1)can someone help to tell how to travel from nagasaki to beppu by train, wiithout going back to hakata again? if need to change train line, how much transit time needed normally?
2)cann we leave luggage bag at the train station?
3)among beppu,yufuin and oita, which is the must to visit?
thanks

1)
via JR Tosu,JR Kumamoto and JR Aso sta
or via JR Yufuin route.
http://www.japan-guide.com/list/e1108.html

2)
Which station?

3)
http://www.japan-guide.com/list/e1243.html
http://www.visit-oita.jp/index.e.html

A scenic way or a faster way?
Scenic route is to pick up the LTD Exp. i.e. Yufuin-no-mori, Yufu, etc. at Tosu that is coming from Hakata. Make sure the train goes all the way to Beppu as some don't and terminate at Yufuin. Or you can pick up a slow going yellow one man local diesel car at Kurume.
If you want to get to Beppu quickly, Go to Hakata or Kokura by shinkansen from Tosu and pick up Sonic LTD Exp. to Beppu.
Check hyperdia.com for the timetable.