While it is definitely not hard to get yourself some fusion cores in the game Fallout 4, it is also nice to know some of the places where you will be able to pick some up in case you run out or you happen to be in that area.
When you complete the third quest of the game, your next goal will be to fo to the Kellogg house. Make your way to the house. Once there, find the desk and crouch down, you should be able to see a red button. Press it and you will find four fusion cores.
Other locations where the fusion cores can be found includes the Castle but you will need to take down the aquatic opponents first. There will be more at Ford Hagen at the western edge of the commonwealth and the generator room. The Weston Water Treatment plant will have more for you.
