I went to Badain years ago. It is best combined with the Zhangye Danxia because of the close proximity - at least for China. These days you can fly into Jinchang and get a taxi from there. From Zhangye I had a driver taking me to the entrance of the park which is 2-3 hours driving so you leave really early or stay in the park which is usually recommended. I didn't because I saw all I wanted within a day. The entrance of the park doesn't really have much to do.

After paying for the ticket (expensive by the way) you can roam around the park but it's still far to trek anywhere so your best bet is to rent a jeep the park offers. It will save you so much time. It can take you through the park to the "thousands of lakes in the desert" although Badain Lake (巴丹湖) is the most popular one which is probably why you need ANOTHER ticket just to enter that area. The main thing to see in my opinion are the various lakes (doesn't have to be Badain Lake), the huuuuuge sand dunes and the singing sand (or singing dunes if you prefer). The famous Bilutu Peak is just in-between the many lakes.

After perhaps 4-5 hours inside the park it was time to go back which means exchanging cars and a long drive back (to Zhangye or Jinchang).

Overall this is an expensive trip because the driver + entrance and you can see similar deserts around the world but I still recommend it. I would probably also recommend a longer trip if you can afford it (time and money).