Hardcore History is great, and his other show Common Sense is fantastic as well. I also like American Life and Radio Lab, but the ones that are a bit less known:
The Last Podcast On The Left is easily my favorite podcast out there. Their themes ranges from serial killers, to hilarious conspiracies. It's all done with great humor and just a good take on things. They do real research but often with a funny angle to it all. I would recommend starting with a heavy hitter like Manson or Dahmer.
The other shows on the network that got last podcast, like roundtable of gentlemen, abe lincons top hat and brighter side are also good.
99% invisible it focuses on design and architecture, which sounds boring, but it's not. Trust me.
Criminal Very good podcast. I wish they would release more frequently and longer episodes, but it's really good.
Designer notes A podcast that is just a bunch of really long in depth interviews with game designers. Sid meier and Amy Hennig ones are particuarly good.
You are not so smart podcast. Basically just goes down a long list of fallacies and other dumb shit you tell yourself.
Sword and Scale can be decent, but the narrator annoys me to death. His constant whining about JUSTICE, and begging for donations, for what is really just him talking a bit of shit and then putting in interesting clips from youtube, is extremely annoying. But he often has some interesting topics and the clips that he eventually gets to, after crying about JUSTICE FOR THE FAMILIES AND VICTIMS, can be really good.