I have three days in Changchun what can I see and where can I go in this short time?

Oh, there are many popular scenic areas in this city.
You can arrange your trip in the following way: On the arrival day, go to Chongqing Road Commercial Street for possible shopping and good dining. For the subsequent three days' itinerary:
Day 1: Jilin Provincial Museum, City Sculpture Art Park
Day 2:Jingyuetan National Forest Park
Day 3:Nanhu Park (South Lake Park), Film City

June is an appropariate time to go sightseeing. You can fully appreciate the beautiful natural scenery. :-)

Based on info above do we have enough time to make a trip to Diaoshuihu Scenic Area ?

Well, this scenic spot is about 48 miles away from the city center. If you want to tour around all the scenic spot mentioned before, you don't have time to visit this one.
You are suggested to spare a whole day to tour around this forest park.