Can I go around Harbin in 3 days without taking a tour?

Of course you can travel on your own. The suggested attractions are Sun Island Scenic Area, Polarland, Siberian Tiger Park, St. Sophia Cathedral and Central Street. If you will go in winter, especially during Jan. - Feb. the Ice and Snow Festival cannot be missed. For that, the Zhaolin Park, and Ice and Snow World are highly recommended as well.

Please, suggest me plan for 3 days in Harbin. Thanks alot.

I presume you will reach in winter. You may refer to the itinerary:
Day 1: Ice and Snow Festival: Central Park - Ice and Snow World - Sun Island Park - Zhaolin Park for Ice Lantern Show
Day 2: Yabuli Ski Resort for a day.
Day 3: St. Sophia Cathedral - Siberian Tiger Park - Polarland

Have a good trip!