Yufuin can be reached the fastest by car from Beppu, taking just 30 minutes. I personally recommend riding the special train Yufu no Mori from Beppu to Yufuin, as it adds to the unique charm that Yufuin has. Once you arrive at Yufuin Station, you will see the streets filled with shops and restaurants catering to all your needs. Souvenir shops and specialty goods shops line both sides of the road, and it's hard to resist the many cafes selling delectable looking sweets and cakes. My suggested plan of action is to walk down this tourist-area road (it's hard to miss as it spreads out straight ahead from the train station) looking at the shops and restaurants on one side until you reach Kinrinko at the end of the road, and then look at the shops and restaurants on the other side of the road As you walk back.
Since Yufuin is a small town, the area can be easily seen within 3 hours making for a great morning or afternoon/evening day trip. If you want to enjoy the onsen at Yufuin, you might want to either spend a night there, or choose to spend the entire day there and leave at night time. The trains are much scarcer running from Yufuin at night time so make sure to check the timetable if you don't plan on staying the night.
At the end of the shops and restaurants road, follow the signs pointing you towards Kinrinko and be rewarded with an absolutely stunning view. The lake is surrounded by mountains that are coloured in beautiful shades of red and gold in autumn time. The still lake water provides gorgeous reflections of the surrounding nature, and a little wooden walkway around the lake gives you all the angles to take in the scenic view. Overall, do make sure to visit Yufuin if you are in the Oita area for some breathtaking views.