The Chernobyl Museum is a must.  Take a taxi and have the address handy, as it is not a very visible buidling.  The address is 1 Kharyvyj Pereulok Street.  Close to the Kontraktova Ploshka (blue line) metro station.  It is also a short walk from Podil .
The museum itself is a quiet building with a handful of emergency vehicles from the time of the disaster parked outside.

 

Once you get in, it is highly recommended that you hire an English speaking guide since everything is in Russian.  The guide can also operate all the multimedia displays in the museum.  Just the movies alone are worth the price of the guide. 

You will be able to get a very accurate and powerful glimpse into the events before, during and after the accident happened.  Perhaps since Ukraine is no longer under Russian rule, they can be quite blunt now about the cover ups, the amazing rescue operations and the many people who gave their lives to stop the catastrophe from getting bigger.

At the end of the tour, you are invited to sit down and watch a movie about what Chernobyl and the surrounding city and villages look like now.  The video shows images of ghost towns.